THE CLASSIFICATION OF 2-COMPACT GROUPS

KASPER K. S. ANDERSEN AND JESPER GRODAL

Abstract. We prove that any connected 2-compact group is classified by its 2-adic root datum, and in particular the exotic 2-compact group DI(4), constructed by Dwyer-Wilkerson, is the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group. Combined with our earlier work with Møller and Viruel for p odd, this establishes the full classification of p-compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p-compact groups and root data over the p-adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen-Grodal-Møller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.

1. Introduction

In this paper we prove that any connected 2-compact group X is classified, up to isomorphism, by its root datum D_{X} over the 2-adic integers ℤ_{2}. This, combined with our previous work with Møller and Viruel [8] for odd primes, finishes the proof of the classification of p-compact groups for all primes p. The classification states that, up to isomorphism, there is a one-to-one correspondence between connected p-compact groups and root data over ℤ_{p}. Hence the classification of p-compact groups is completely parallel to the classification of compact connected Lie groups [9, §4, no. 9], just with ℤ replaced by ℤ_{p}. The classification of 2-compact groups has the following consequence, which captures most of the classification statement.

Theorem 1.1 (Classification of 2-compact groups; splitting version). Let X be a connected 2-compact group. Then BX ≃ BG_{ˆ2} × B DI(4)^{s}, s ≥ 0, where G is a compact connected Lie group, and B DI(4) is the classifying space of the exotic 2-compact group DI(4) constructed by Dwyer-Wilkerson [24].

This is the traditional form of the classification conjecture e.g., stated by Dwyer in his 1998 ICM address as [19, Conjs. 5.1 and 5.2]. A p-compact group, introduced by Dwyer-Wilkerson [25], can be defined as a pointed, connected, p-complete space BX with H^{∗}(ΩBX; 𝔽_{p}) finite over 𝔽_{p}, and X is then the pointed loop space ΩBX. The space BX is hence the classifying space of the loop space X, justifying the convention of referring to the p-compact group simply by X. A p-compact group is called connected if the space X is connected, and two p-compact groups are said to be isomorphic if their classifying spaces are homotopy equivalent. For more background on p-compact groups, including details on the history of the classification conjecture, we refer to [8] and the references therein—we also return to it later in this introduction.

The second-named author was partially supported by NSF grant DMS-0354633 and an Alfred P. Sloan Research Fellowship.

To make the classification more precise, we now recall the notion of a root datum over ℤ_{p}. For p = 2 this theory provides a key new input to our proofs, and was developed in the paper [28] of Dwyer-Wilkerson and in our paper [4] (see also Section 8); we say more about this later in the introduction where we give an outline of the proof of Theorem 1.2.

For a principal ideal domain R, an R-root datum D is a triple (W, L, {Rb_{σ}}), where L is a finitely generated free R-module, W ⊆ Aut_{R}(L) is a finite subgroup generated by reflections (i.e., elements σ such that 1−σ ∈ End_{R}(L) has rank one), and {Rb_{σ}} is a collection of rank one submodules of L, indexed by the reflections σ in W, satisfying the two conditions

im(1 − σ) ⊆ Rb_{σ} and w(Rb_{σ}) = Rb_{wσw^{−1}} for all w ∈ W.

The element b_{σ} ∈ L, determined up to a unit in R, is called the coroot corresponding to σ, and together with σ, it determines a root β_{σ} : L → R via the formula σ(x) = x + β_{σ}(x)b_{σ}. There is a one-to-one correspondence between ℤ-root data and classically defined root data, by associating (L, L^{∗}, {±b_{σ}}, {±β_{σ}}) to (W, L, {ℤb_{σ}}); see [28, Prop. 2.16]. For both R = ℤ or ℤ_{p}, one can, instead of {Rb_{σ}}, equivalently consider their span, the coroot lattice, L_{0} = +_{σ}Rb_{σ} ⊆ L, the definition given in [8, §1] (under the name “R-reflection datum”). For R = ℤ_{p}, p odd, the notion of an R-root datum agrees with that of an R-reflection group (W, L); see Section 8. Given two R-root data D = (W, L, {Rb_{σ}}) and D′ = (W′, L′, {Rb′_{σ}}), an isomorphism between D and D′ is an isomorphism φ : L → L′ such that φW φ^{−1} = W′ as subgroups of Aut(L′) and φ(Rb_{σ}) = Rb′_{φσφ^{−1}} for every reflection σ ∈ W. We let Aut(D) be the automorphism group of D, and we define the outer automorphism group as Out(D) = Aut(D)/W. A classification of ℤ_{p}-root data is given as Theorems 8.1 and 8.12.

We now explain how to associate a ℤ_{p}-root datum to a connected p-compact group. By a theorem of Dwyer-Wilkerson [25, Thm. 8.13] any p-compact group X has a maximal torus, which is a map i : BT = (BS^{1}_{p̂})^{r} → BX satisfying that the homotopy fiber has finite 𝔽_{p}-cohomology and non-trivial Euler characteristic. Replacing i by an equivalent fibration, we define the Weyl space W_{X}(T) as the topological monoid of self-maps BT → BT over i. The Weyl group is defined as W_{X}(T) = π_{0}(W_{X}(T)) and the classifying space of the maximal torus normalizer is defined as the Borel construction BN_{X}(T) = BT_{hW_{X}(T)}. By definition, W_{X} acts on L_{X} = π_{2}(BT) and if X is connected, this gives a faithful representation of W_{X} on L_{X} as a finite ℤ_{p}-reflection group [25, Thm. 9.7(ii)]. There is also an easy formula for the coroots b_{σ} in terms of the maximal torus normalizer N_{X}, for which we refer to Section 8.1 (or [28], [4]). We define D_{X} = (W_{X}, L_{X}, {ℤ_{p}b_{σ}}).

We are now ready to state the precise version of our main theorem.

Theorem 1.2 (Classification of p-compact groups). The assignment which to a connected p-compact group X associates its ℤ_{p}-root datum D_{X} gives a one-to-one correspondence between connected p-compact groups, up to isomorphism, and ℤ_{p}-root data, up to isomorphism. Furthermore the map Φ : Out(BX) → Out(D_{X}) is an isomorphism and more generally

B Aut(BX) ⟶^{≃} (B^{2}Z(D_{X}))_{h Out(D_{X})}

where the action of Out(D_{X}) on B^{2}Z(D_{X}) is the canonical one.

Here Aut(BX) is the space of self-homotopy equivalences of BX, Out(BX) is its component group, B^{2}Z(D_{X}) is the double classifying space of the center of the ℤ_{p}-root datum D_{X} (see Proposition 8.4(1)) and Φ is the standard Adams-Mahmud map [8, Lem. 4.1] given by lifting a self-equivalence BX to BT, see Recollection 8.2. We remark that the existence of the map B Aut(BX) → (B^{2}Z(D_{X}))_{h Out(D_{X})} in the last part of the theorem, requires the knowledge that the fibration B Aut(BX) → B Out(BX) splits, which was established in [4, Thm. C]. Theorem 1.2 implies that connected p-compact groups are classified by their maximal torus normalizer, the classification conjecture in [19, Conj. 5.3]. For p odd, Theorem 1.2 is [8, Thm. 1.1] (with an improved description of B Aut(BX) by [4, Thm. C]). Our proof here is written so that it is independent of the prime p—see the outline of proof later in this introduction for a further discussion.

The main theorem has a number of important corollaries. The “maximal torus conjecture”, gives a purely homotopy theoretic characterization of compact Lie groups amongst finite loop spaces:

Theorem 1.3 (Maximal torus conjecture). The classifying space functor, which to a compact Lie group G associates the finite loop space (G, BG, e : G ⟶^{≃} ΩBG) gives a one-to-one correspondence between compact Lie groups and finite loop spaces with a maximal torus.

Furthermore, for G connected, B Aut(BG) ≃ (B^{2}Z(G))_{h Out(G)}.

The automorphism statement above is included for completeness, but follows easily by combining previous work of Jackowski-McClure-Oliver, Dwyer-Wilkerson, and de Siebenthal (cf., [33, Cor. 3.7], [26, Thm. 1.4], and [15, Ch. I, §2, no. 2]). The maximal torus conjecture seems to first have made it into print in 1974, where Wilkerson [65] described it as a “popular conjecture towards which the author is biased”.

The “Steenrod problem” from around 1960 (see Steenrod’s papers [57, 56]), asks which graded polynomial algebras are realized as the polynomial ring of some space X? The problem was solved with 𝔽_{p}-coefficients, for p “large enough”, by Adams-Wilkerson [2] in 1980, extending work of Clark-Ewing [13], and for all odd p, by Notbohm [51] in 1999. The case p = 2 is different from odd primes, for instance since generators can appear in odd degrees.

Theorem 1.4 (Steenrod’s problem for 𝔽_{2}). Suppose that P^{∗} is a graded polynomial algebra over 𝔽_{2} in finitely many variables. If H^{∗}(Y; 𝔽_{2}) ≅ P^{∗} for some space Y, then P^{∗} is isomorphic, as a graded algebra, to

H^{∗}(BG; 𝔽_{2}) ⊗ H^{∗}(B DI(4); 𝔽_{2})^{⊗r} ⊗ H^{∗}(ℝ P^{∞}; 𝔽_{2})^{⊗s} ⊗ H^{∗}(ℂ P^{∞}; 𝔽_{2})^{⊗t}

for some r, s, t ≥ 0, where G is a compact connected Lie group with finite center and ℝ P^{∞} and ℂ P^{∞} denotes infinite dimensional real and complex projective space, respectively. In particular if P^{∗} has all generators in degree ≥ 3 then P^{∗} is a tensor product of the following graded algebras:

H^{∗}(B SU(n); 𝔽_{2}) ≅ 𝔽_{2}[x_{4}, x_{6}, …, x_{2n}],
H^{∗}(B Sp(n); 𝔽_{2}) ≅ 𝔽_{2}[x_{4}, x_{8}, …, x_{4n}],
H^{∗}(B Spin(7); 𝔽_{2}) ≅ 𝔽_{2}[x_{4}, x_{6}, x_{7}, x_{8}],
H^{∗}(B Spin(8); 𝔽_{2}) ≅ 𝔽_{2}[x_{4}, x_{6}, x_{7}, x_{8}, y_{8}],
H^{∗}(B Spin(9); 𝔽_{2}) ≅ 𝔽_{2}[x_{4}, x_{6}, x_{7}, x_{8}, x_{16}],
H^{∗}(BG_{2}; 𝔽_{2}) ≅ 𝔽_{2}[x_{4}, x_{6}, x_{7}],
H^{∗}(BF_{4}; 𝔽_{2}) ≅ 𝔽_{2}[x_{4}, x_{6}, x_{7}, x_{16}, x_{24}],
H^{∗}(B DI(4); 𝔽_{2}) ≅ 𝔽_{2}[x_{8}, x_{12}, x_{14}, x_{15}].

Since the classification of p-compact groups is a space-level statement, it also gives which graded polynomial algebras over the Steenrod algebra can occur as the cohomology rings of a space; e.g., the decomposition in Theorem 1.4 where P^{∗} is assumed to have generators in degrees ≥ 3 also holds over the Steenrod algebra. It should also be possible to give a more concrete list even without the degree ≥ 3 assumption, by finding all polynomial rings which occur as H^{∗}(BG; 𝔽_{2}) for G a compact connected Lie group with finite center; for G simple a list can be found in [36, Thm. 5.2], cf. Remark 7.1. In a short companion paper [6] we show how the theory of p-compact groups in fact allows for a solution to the Steenrod problem with coefficients in an arbitrary commutative ring R.

We can also determine to which extent the realizing space is unique: Recall that two spaces Y and Y′ are said to be 𝔽_{p}-equivalent if there exists a space Y′′ and a zig-zag Y → Y′′ ← Y′ inducing isomorphisms on 𝔽_{p}-homology. The statement below also holds verbatim when p is odd, where the result is due to Notbohm [51, Cors. 1.7 and 1.8]—again complications arise for p = 2, e.g., due to the possibility of generators in odd degrees.

Theorem 1.5 (Uniqueness of spaces with polynomial 𝔽_{2}-cohomology). If A^{∗} is a graded polynomial 𝔽_{2}-algebra over the Steenrod algebra A_{2}, in finitely many variables, all in degrees ≥ 3, then there exists at most one space Y, up to 𝔽_{2}-equivalence, with H^{∗}(Y; 𝔽_{2}) ≅ A^{∗}, as graded 𝔽_{2}-algebras over the Steenrod algebra.

If P^{∗} is a finitely generated graded polynomial 𝔽_{2}-algebra, then there exists at most finitely many spaces Y up to 𝔽_{2}-equivalence such that H^{∗}(Y; 𝔽_{2}) ≅ P^{∗} as graded 𝔽_{2}-algebras.

The early uniqueness results on p-compact groups starting with [21], which predate root data, or even the formal definition of a p-compact group, were formulated in this language— we give a list of earlier classification results later in the introduction. The assumption that all generators are in degrees ≥ 3 for the first statement cannot be dropped since for instance B(S^{1} × SU(p^{2})) and B((S^{1} × SU(p^{2}))/C_{p}) have isomorphic 𝔽_{p}-cohomology algebras over the Steenrod algebra, but are not 𝔽_{p}-equivalent. Also, the same graded polynomial 𝔽_{p}- algebra can of course often have multiple Steenrod algebra structures, the option left open in the second statement: B SU(2) × B SU(4) and B Sp(2) × B SU(3) have isomorphic 𝔽_{p}- cohomology algebras, but with different Steenrod algebra structures at all primes.

Bott’s theorem on the cohomology of X/T, the Peter-Weyl Theorem, and Borel’s characterization of when centralizers of elements of order p are connected, given for p odd as Theorems 1.5, 1.6, and 1.9 of the paper [8], also hold verbatim for p = 2 as a direct consequence of the classification (see Remark 7.3). Likewise [8, Thm. 1.8], giving different formulations of being p-torsion free, holds verbatim except that condition (3) should be removed, cf. [8, Rem. 10.10]. We also remark that the classification together with results of Bott for compact Lie groups, gives that H^{∗}(ΩX; ℤ_{p}) is p-torsion free and concentrated in even degrees for all p-compact groups. This result was first proved by Lin and Kane, in fact in the more general setting of finite mod p H-spaces, in a series of celebrated, but highly technical, papers [38, 39, 40, 35], using completely different arguments.

Theorem 1.2 also implies a classification for non-connected p-compact groups, though, just as for compact Lie groups, the classification is less calculationally explicit: Any disconnected p-compact group X fits into a fibration sequence

BX_{1} → BX → Bπ

with X_{1} connected, and since our main theorem also includes an identification of the classifying space of such a fibration B Aut(BX_{1}) with the algebraically defined space (B^{2}Z(D_{X_{1}}))_{h Out(D_{X_{1}})}, this allows for a description of the moduli space of p-compact groups with component group π and whose identity component has ℤ_{p}-root datum D. More precisely we have the following theorem, which in the case where π is the trivial group recovers our classification theorem in the connected case.

Theorem 1.6 (Classification of non-connected p-compact groups). Let D be a ℤ_{p}-root datum, π a finite p-group and set B aut(D) = (B^{2}Z(D))_{h Out(D)}. The space

M = (map(Bπ, B aut(D)))_{h Aut(Bπ)}

classifies p-compact groups whose identity component has ℤ_{p}-root datum isomorphic to D and component group isomorphic to π, in the following sense:

(1) There is a one-to-one correspondence between isomorphism classes of p-compact groups X with π_{0}(X) ≅ π and D_{X_{1}} ≅ D, and components of M, given by associating to X the component of M given by the classifying map Bπ → B Aut(BX_{1}) ⟶^{≃} B aut(D_{X_{1}}). In particular the set of isomorphism classes of such p-compact groups identifies with the set of Out(π)-orbits on [Bπ, B aut(D)], which is finite.

(2) For each p-compact group X the corresponding component of M has the homotopy type of B Aut(BX) via the zig-zag

B Aut(BX) ⟵^{≃} (map(Bπ, B Aut(BX_{1}))_{C(f)})_{h Aut(Bπ)} ⟶^{≃} (map(Bπ, B aut(D_{X_{1}}))_{C(f)})_{h Aut(Bπ)}

where C(f) denote the Out(π)-orbit on [Bπ, B Aut(BX_{1})] of the element classifying f : BX → Bπ.

Finally, remark that the uniqueness part of the classification Theorem 1.2 can be reformulated as an isomorphism theorem stating that the isomorphisms, up to conjugation, between two arbitrary connected p-compact groups are exactly the isomorphisms, up to conjugation, between their root data. For algebraic groups the isomorphism theorem can be strengthened to an isogeny theorem stating that isogenies of algebraic groups correspond to isogenies of root data; see e.g., [60]. In another companion paper [5] we deduce from our classification theorem that the same is true for p-compact groups: Homotopy classes of maps BX → BX′ which induce isomorphism in rational cohomology (the notion of an isogeny for p-compact groups) are in one-to-one correspondence with the conjugacy classes of isogenies between the associated ℤ_{p}-root data, sending isomorphisms to isomorphisms. Here an isogeny of ℤ_{p}- root data (W, L, {ℤ_{p}b_{σ}}) → (W′, L′, {ℤ_{p}b′_{σ}}) is a ℤ_{p}-linear monomorphism φ : L → L′ with finite cokernel, such that the induced isomorphism φ : Aut(L ⊗_{ℤ_{p}} ℚ_{p}) → Aut(L′ ⊗_{ℤ_{p}} ℚ_{p}) sends W isomorphically to W′ and such that φ(ℤ_{p}b_{σ}) = ℤ_{p}b′_{φ(σ)}, for every reflection σ ∈ W where the corresponding factor of W has order divisible by p. As a special case this theorem also contains the description of rational self-equivalences of p-completed classifying spaces of compact connected Lie groups, the most general of the theorems obtained by Jackowski-McClure-Oliver in [33], illuminating their result.

Structure of the paper and outline of the proof of the classification. Our proof of the classification of 2-compact groups, written to work for any prime, follows the same overall structure as our proof for p odd with Møller and Viruel in [8], but with significant additions and modifications. Most importantly we draw on the theory of root data [28] [4] and have a different way of dealing with the obstruction group problem. We outline our strategy below, and also refer the reader to [8, Sec. 1] where we discuss the proof for p odd.

An inspection of the classification of ℤ_{p}-root data, Theorem 8.1, shows that all ℤ_{p}-root data have already been realized as root data of p-compact groups by previous work, so only uniqueness is an issue. (For p = 2 the root datum of DI(4) [24] is the only irreducible ℤ_{2}-root datum not coming from a ℤ-root datum; for p odd see [8].)

The proof that two connected p-compact groups with isomorphic ℤ_{p}-root data are isomorphic, is divided into a prestep and three steps, spanning Sections 2–6. Before describing these steps, we have to recall some necessary results about root data and maximal torus normalizers from [28] and [4]: The first thing to show is that the maximal torus normalizer N_{X} can be explicitly constructed from D_{X}. For p odd this follows by a theorem of the first-named author [3] stating that the maximal torus normalizer N_{X} is always split (i.e., the fibration BT → BN_{X} → BW_{X} splits). For p = 2 this is not necessarily the case, and the situation is more subtle: Recently Dwyer-Wilkerson [28] showed how to extend part of the classical paper of Tits [61] to p-compact groups, in particular reconstructing N_{X} from D_{X}. Since the automorphisms of N_{X} differ from those of X, one however for classification purposes has to consider an additional piece of data, namely certain “root subgroups” {N_{σ}}, which one can define algebraically for each reflection σ. We construct these for pcompact groups in [4], see also Section 8, and describe a candidate algebraic model for the space B Aut(BX). (In the setting of algebraic groups, this “root subgroup” N_{σ} will be the maximal torus normalizer of ⟨U_{α}, U_{−α}⟩, where U_{α} is the root subgroup in the sense of algebraic groups corresponding to the root α dual to the coroot b_{σ}; see [4, Rem. 3.8].) For the connoisseur we note that the reliance in [28] on a classification of connected 2-compact groups of rank 2 was eliminated in [4, version 2].

In [4], recalled in Recollection 8.2, we show that the “Adams-Mahmud” map, which to a homotopy equivalence of BX associates a homotopy equivalence of BN_{X}, factors

Φ : Out(BX) → Out(BN , {BN_{σ}}) ⟶^{≅} Out(D_{X})

where Out(BN , {BN_{σ}}) is the set of homotopy classes of self-homotopy equivalences of BN_{X} permuting the root subgroups BN_{σ} (see Recollection 8.2 for the precise definition). As a part of our proof we will show that Φ is a isomorphism by induction.

The main argument proceeds by induction on the cohomological dimension of X, which can be determined from (W_{X}, L_{X}) alone. (Almost equivalently one could do induction on the order of W_{X}.) It is divided into a prestep (Section 2) and three steps (Sections 4–6). Prestep (Section 2): The first step is to reduce to the case of simple, center-free groups. For this we use rather general arguments with fibrations and their automorphisms, in spirit similar to the arguments in [8]. However the theory of root data and root subgroups is needed both for the statements and results for p = 2, and these are incorporated throughout.

With this in hand, we can assume that we have two connected simple, center-free pcompact groups X and X′ with isomorphic ℤ_{p}-root data D_{X} and D_{X′}. As explained in the discussion above, [28, 4] implies that the corresponding maximal torus normalizers and root subgroups are isomorphic, and we can hence assume that they both equal (BN , {BN_{σ}}) embedded via maps j and j′ in X and X′,

\xymatrix{ & (BN , \{BN_{σ}\}) \ar[dr]^{j′} \ar[dl]_{j} \\ BX \ar@{.>}[rr] &  & BX′}

where the dotted arrow is the one that we want to construct.

Step 1 (Section 4): Using that in a connected p-compact group every element of order p can be conjugated into the maximal torus, uniquely up to conjugation in N, we have, for every element ν : Bℤ/p → BX, of order p in X, a diagram of the form

\xymatrix{ & (BC_{N} (ν), \{BC_{N} (ν)_{σ}\}) \ar[dr] \ar[dl] \\ BC_{X}(ν) &  & BC_{X′}(ν).}

We can furthermore take covers of this diagram with respect to the fundamental group π_{1}(D) of the root datum, which we indicate by adding a tilde ˜(·). (This uses the formula for the fundamental group of a p-compact group [29], but see also Theorem 8.6.) In Section 4 we prove that one can use the induction hypothesis to construct a homotopy equivalence between B{ \atop C_{X}(ν)} and B{ \atop C_{X′}(ν)} under B{ \atop C_{N} (ν)}. The tricky point here is that these centralizers need not themselves be connected, so one first has to construct the map on the identity component BC_{X}(ν)_{1} and then show that it extends, and this in turn requires that one has control of the space of self-equivalences of BC_{X}(ν)_{1}.

Now for a general elementary abelian p-subgroup ν : BE → BX of X we can pick an element of order p in E, and restriction provides a map

B{ \atop C_{X}(ν)} → B{ \atop C_{X}(ℤ/p)} → B{ \atop C_{X′}(ℤ/p)} → BX̃′.

Step 2 (Section 5): To make sure that these maps are chosen in a compatible way, one has to show that this map does not depend on the choice of rank one subgroup of E. In Section 5 we show that this lift does not depend on the choices, relying on techniques developed in [8]. We furthermore show that they combine to form an element

[ϑ] ∈ lim _{ν∈A(X)}^{0}[B{ \atop C_{X}(ν)}, BX̃′]

where A(X) is the Quillen category of X, with objects the elementary abelian p-subgroups of X and morphisms induced by conjugation in X. Step 3 (Section 6): The construction of the element [ϑ] basically guarantees that X and X′ have the same p-fusion, and the last step, which we carry out in Section 6, deals with the rigidification question, where our approach differs significantly from [8]. In particular, since we work on universal covers throughout, we are able to relate our obstruction groups to groups already calculated in [32]. Since the exotic p-compact groups (only DI(4) for p = 2) are easily dealt with, we can assume that BX = BG_{p̂} for some simple, center-free Lie group. Jackowski-McClure-Oliver showed in [32] that BG_{p̂} can be expressed as a homotopy colimit of certain subgroups P of G, the so-called p-radical (also known as p-stubborn) subgroups. For a p-radical subgroup P of G, our element [ϑ] above gives maps

BP̃_{p̂} → B{ \atop C_{G}(_{p}Z(P))}_{p̂} → BX̃′,

where _{p}Z(P) denotes the subgroup of elements of order at most p in Z(P). These maps combine to form an element in

lim _{G̃/P̃∈O_{p}^{r}(G̃)}^{0}[BP̃_{p̂}, BX̃′]

where O_{p}^{r}(G̃) is the full subcategory of the p-orbit category of G̃ with objects G̃/P̃ for P̃ a p-radical subgroup of G̃.

The obstructions to rigidifying this to get a map on the homotopy colimit

hocolim_{G̃/P̃∈O_{p}^{r}(G̃)}(EG̃ ×G̃ G̃/P̃)_{p̂} → BX̃′

lies in obstruction groups which identify with

lim _{G̃/P̃∈O_{p}^{r}(G̃)}^{∗}π_{∗}(Z(P̃)_{p̂}).

Using extensive case-by-case calculations, Jackowski-McClure-Oliver showed in [32] that these obstructions in fact vanish. Hence we have constructed a map

BG̃_{p̂} ⟵^{≃} (hocolim_{G̃/P̃∈O_{p}^{r}(G̃)}(EG̃ ×G̃ G̃/P̃)_{p̂})_{p̂} → BX̃′

which is easily seen to be an equivalence. Passing to a quotient we get the wanted equivalence BG_{p̂} → BX′, finishing the proof of uniqueness. The remaining statements of Theorem 1.2 also fall out of this approach. Section 7 proves the stated consequences of the classification and the appendix Section 8 is used to establish a number of general properties of root data over ℤ_{p} used throughout the paper.

Here is the contents in table form:

Contents

1. Introduction 1 2. Reduction to center-free simple p-compact groups 9 3. Preliminaries on self-equivalences of non-connected p-compact groups 13 4. First part of the proof of Theorem 1.2: Maps on centralizers 15 5. Second part of the proof of Theorem 1.2: The element in lim^{0} 18 6. Third and final part of the proof of Theorem 1.2: Rigidification 24 7. Proof of the corollaries of Theorem 1.2 29 8. Appendix: Properties of ℤ_{p}-root data 34 8.1. Root datum, normalizer extension, and root subgroups of a p-compact group 35 8.2. Centers and fundamental groups 36 8.3. Covers and quotients 38 8.4. Automorphisms 42 8.5. Finiteness properties 43 References 44

Related work and acknowledgments. We refer to the introduction of our paper [8] with Møller and Viruel for a detailed discussion of the history of the classification for odd primes. The first classification results for 2-compact groups were obtained by Dwyer-Miller-Wilkerson [21] twenty years ago, in the fundamental cases SU(2) and SO(3). Notbohm [50] and Møller-Notbohm [46] covered SU(n). Viruel covered G_{2} [64], Viruel-Vavpetič covered F_{4} and Sp(n) [63] [62], Morgenroth and Notbohm handled SO(2n+1) and Spin(2n+1) [47] [52], and Notbohm proved the result for DI(4) [53], all using arguments specific to the case in question.

Obviously this paper owes a great debt to our earlier work with Møller and Viruel [8] for odd primes. Jesper Møller introduced us to the induction-on-centralizers approach to the classification, and Antonio Viruel gave us the idea of trying to compare the centralizer and p-stubborn decomposition, a method he had used in the paper [63] in a special case. We are very grateful to them for sharing their insights. We would furthermore like to thank Bill Dwyer and Clarence Wilkerson for helpful correspondence, and for sharing an early version of their manuscript [29] on fundamental groups of p-compact groups with us, and Haynes Miller for useful questions. The results of this paper were announced in Spring 2005 e.g., at the Conference on Topology at the Isle of Skye, June 2005 [7]. Independently of our results, Jesper Møller has announced a proof of the classification of connected 2-compact groups (Theorem 1.1) using computer algebra [41]. We benefited from the hospitality of Aarhus University, University of Copenhagen, and University of Chicago while writing this paper.

2. Reduction to center-free simple p-compact groups

In this section we reduce the classification of connected p-compact groups to the case of simple center-free groups, in the sense that the classification statement, Theorem 1.2, holds for a connected p-compact group if it holds for the simple factors occurring in the corresponding adjoint (center-free) p-compact group (see Propositions 2.1 and 2.4). We do this by extending the proofs given in [8, Sec. 6] for p odd, to all primes by incorporating ℤ_{p}-root data, and root subgroups. Since this additional data requires restructuring of most of the proofs, we present this reduction in some detail.

As in [8] we make the following working definition: A connected p-compact group X is said to be determined by its ℤ_{p}-root datum D_{X} if any connected p-compact group X′ with D_{X′} ≅ D_{X} is isomorphic to X. (Theorem 1.2 will eventually show that this always holds.)

Proposition 2.1 (Product Reduction). Suppose X = X_{1} × ⋯ × X_{k} is a product of simple p-compact groups.

(1) If Φ : Out(BX_{i}) → Out(D_{X_{i}}) is injective for each i, then so is Φ : Out(BX) → Out(D_{X}).

(2) If Φ : Out(BX_{i}) → Out(D_{X_{i}}) is surjective and X_{i} is determined by D_{X_{i}} for each i, then Φ : Out(BX) → Out(D_{X}) is surjective.

Proof. The proof of (1) is identical to the proof of [8, Lem. 6.1(2)] (the key fact is knowing that a map out of a connected p-compact which is trivial when restricted to the maximal torus is in fact trivial, which e.g., follows from [42, Thm. 6.1]). The statement in (2) is a direct consequence of the description of Out(D) in Proposition 8.13 together with the assumption that if D_{X_{i}} is isomorphic to D_{X_{j}} then X_{i} is isomorphic to X_{j}. □

The reader might want to note that conversely to Proposition 2.1(2), if Φ : Out(BX) → Out(D_{X}) is surjective for all connected p-compact groups X, then all connected p-compact groups are determined by their root datum, as is seen by considering products.

Construction 2.2 (Quotients of p-compact groups). For explicitness we recall the quotient construction for p-compact groups, and describe when a self-homotopy equivalence induces a homotopy equivalence on quotients, since this will be used in what follows:

Let X be a p-compact group, A an abelian p-compact group and i : BA → BX a central homomorphism. By assumption BA is homotopy equivalent to map(BA, BA)_{1} and map(BA, BX)_{i} ⟶^{ev} BX is an equivalence. Recall that the quotient BX/A is defined as the Borel construction of the composition action of map(BA, BA)_{1} on map(BA, BX)_{i}, cf. [25, Pf. of Prop. 8.3]. This action and the resulting quotient space BX/A only depends on the (free) homotopy class of i, even on the point-set level, and we have a canonical quotient map q : BX ⟵^{≃} map(BA, BX)_{i} → BX/A, well defined up to homotopy.

Now suppose we have a self-homotopy equivalence f : BX → BX such that there exists a homotopy equivalence g : BA → BA making the diagram

\xymatrix{BA \ar[r]^{g} \ar[d]_{i} & BA \ar[d]^{i} \\ BX \ar[r]^{f} & BX}

commute up to homotopy. We claim that f naturally induces a map on quotients:

First, by using the bar construction model for BA, we can without restriction assume that g is induced by a group homomorphism and has a strict inverse g^{−1}. Next note that in general, if φ : G → G′ is a map of monoids, h : Y → Y′ is a map from a G-space Y to a G′-space Y′, which is φ-equivariant in the sense that h(g · y) = φ(g) · h(y), then there is a canonical induced map on Borel constructions Y_{hG} → Y′_{hG′} under h : Y → Y′ and over Bφ : BG → BG′, e.g., by viewing the Borel construction as a homotopy colimit via the one-sided bar construction.

In the above setup take φ = c_{g}, the monoid automorphism of map(BA, BA)_{1} given by c_{g}(α) = g ∘ α ∘ g^{−1}, and h the self-map of map(BA, BX)_{i} given by β ↦ f ∘ β ∘ g^{−1}. Then f induces a map f̄: BX/A → BX/A, which fits into the homotopy commutative diagram

\xymatrix{BX \ar[d]_{q} \ar[r]^{f} & BX \ar[d]^{q} \\ BX/A \ar[r]^{\overline{f}} & BX/A.}

The quotient construction furthermore behaves naturally with respect to the maximal torus: If j : BT → BX is a maximal torus of X, and h : BT → BT is a lifting of f : BX → BX, then i : BA → BX factors through j and g : BA → BA lifts h [26, Lem. 6.5], and the above construction produces a homotopy commutative diagram

\xymatrix{BT/A \ar[r]^{\overline{h}} \ar[d]_{j/A} & BT/A \ar[d]^{j/A} \\ BX/A \ar[r]^{\overline{f}} & BX/A}

up to homotopy under the diagram one has before taking quotients.

Lemma 2.3. Let X be a connected p-compact group with center i : BZ → BX and let q : BX → BX/Z denote the quotient map. Then any self-homotopy equivalence φ : BX → BX fitting into a homotopy commutative diagram

\xymatrix{ & BZ \ar[dr]^{i} \ar[dl]_{i} \\ BX \ar[dr]_{q} \ar[rr]^{φ} &  & BX \ar[dl]^{q} \\  & BX/Z}

is homotopic to the identity.

Proof. Let q : BX → BX/Z denote the quotient map, turned into a fibration. By changing φ up to homotopy, we can assume that φ is a map strictly over q. By [20] (see also [26, Prop. 11.9]) the homotopy class of φ as a map over q corresponds to an element [φ] ∈ π_{1}(map(BX/Z, B Aut(BZ))_{f}), where f : BX/Z → B Aut(BZ) is the map classifying the fibration q. Note that the class [φ] could a priori depend on how we choose φ, although this turns out not to be the case.

The composite f̄: BX/Z → B Aut(BZ) → B Out(BZ) is null-homotopic since π_{1}(BX/Z) = 0, and obviously map(BX/Z, B Out(BZ))_{0} ⟶^{ev} B Out(BZ) is a homotopy equivalence. We thus have a fibration sequence

map(BX/Z, B Aut_{1}(BZ))_{[f]} → map(BX/Z, B Aut(BZ))_{f} → B Out(BZ)

where [f] denotes the components which map to the component of f. Since B Aut_{1}(BZ) ≃ B^{2}Z is a loop space,

π_{1}(map(BX/Z, B Aut_{1}(BZ))_{g}) ≅ π_{1}(map(BX/Z, B Aut_{1}(BZ))_{0}) = [BX/Z, BZ] = 0,

for any g : BX/Z → B Aut_{1}(BZ), using that BX/Z is simply connected and π_{2}(BX/Z) is finite. Hence π_{1}(map(BX/Z, B Aut(BZ))_{f}) → Out(BZ) is injective. But, since [φ] by assumption maps to the identity in Out(BZ), we conclude that [φ] is the identity, and in particular φ is homotopic to the identity as wanted. □

Proposition 2.4 (Reduction to center-free case). Let X be a connected p-compact group with center Z.

(1) If X/Z is determined by D_{X/Z} and Φ : Out(BX/Z) → Out(D_{X/Z}) is surjective, then X is determined by D_{X}.

(2) If Φ : Out(BX/Z) → Out(D_{X/Z}) is injective then so is Φ : Out(BX) → Out(D_{X}).

(3) If Φ : Out(BX/Z) → Out(D_{X/Z}) is surjective, then so is Φ : Out(BX) → Out(D_{X}).

Proof. The proof of (1) follows the outline of the corresponding statement for odd primes [8, Lem. 6.8(1)], but with the important additional input that we need to keep track of the root subgroups: Suppose that X and X′ are connected p-compact groups with the same ℤ_{p}-root datum D. By [28, Prop. 1.10] X and X′ have isomorphic maximal torus normalizers N_{D}, cf. Section 8. By [4, Thm. 3.1(2)] we can choose monomorphisms j : BN_{D} → BX and j′ : BN_{D} → BX′ such that the root subgroups in BN_{D} with respect to j and j′ agree. Furthermore, the centers of X and X′ agree, and can be viewed as a subgroup Z of N_{D}. Now, N_{D}/Z will be a maximal torus normalizer for both X/Z and X′/Z (see e.g., [44, Thm. 1.2]) and the root subgroups in BN̆_{D}/Z̆ with respect to j/Z and j′/Z also agree by construction. By our assumptions there hence exists a homotopy equivalence f : BX/Z → BX′/Z such that

\xymatrix{ & BN_{D}/Z \ar[dl]_{j/Z} \ar[dr]^{j′/Z} \\ BX/Z \ar[rr]^{f} &  & BX′/Z}

commutes up to homotopy. We need to see that f is a map over B^{2}Z, since this implies that f induces a homotopy equivalence BX → BX′ as wanted. This is a short argument given as the last part of [8, Pf. of Lem. 6.8(1)].

To see (2) consider φ ∈ Aut(BX) corresponding to an element in the kernel of Φ : Out(BX) → Out(D_{X}). By definition, cf. Recollection 8.2, this means that if i : BT → BX is a maximal torus then the diagram

\xymatrix{ & BT \ar[dl]_{i} \ar[dr]^{i} \\ BX \ar[rr]^{φ} &  & BX}

commutes up to homotopy. Since the center BZ → BX factors through i : BT → BX [26, Thm. 1.2], Construction 2.2 combined with our assumption shows that we get a homotopy commutative diagram

\xymatrix{ & BZ \ar[dr] \ar[dl] \\ BX \ar[dr]_{q} \ar[rr]^{φ} &  & BX \ar[dl]^{q} \\  & BX/Z}

so Lemma 2.3 gives the desired conclusion.

We now embark on showing (3), i.e., that Φ : Out(BX) → Out(D_{X}) is surjective, which requires some preparation. Recall that for any connected p-compact group Y, Ỹ is the pcompact group whose classifying space BỸ is the fiber of the fibration BY ⟶^{q} P_{2}BY, where P_{2}BY is the second Postnikov section. Let i : BT → BY be a maximal torus, which we can assume to be a fibration, and let BT̃ denote the fiber of the fibration q ∘ i : BT → P_{2}BY. Since π_{2}(i) : π_{2}(BT) → π_{2}(BY) is surjective (see Proposition 8.5), the long exact sequence of homotopy groups shows that BT̃ is a p-compact torus, and furthermore BT̃ → BỸ is a maximal torus by the diagram

[figure: Ỹ/T̃; BT̃; BỸ; Y/T; BT; i; q∘i; BY; q; ∗; P_{2}BY; P_{2}BY.]

Any self-homotopy equivalence f : BY → BY lifts to a self-homotopy equivalence f̃: BỸ → BỸ, by taking fibers, and it is clear that the assignment f ↦ f̃ induces a homomorphism Out(BY) → Out(BỸ).

For a Y which satisfies π_{1}(D_{Y}) ⟶^{≅} π_{1}(Y), Proposition 8.9 shows that DỸ ≅ D̃_{Y}. Hence, the Adams-Mahmud map Φ, recalled in Recollection 8.2, together with Proposition 8.14 provides the maps in the following diagram

(2.1)  \xymatrix{Out(BY) \ar[d] \ar[r]^{Φ} & Out(D_{Y}) \ar[d] \\ Out(BỸ) \ar[r]^{Φ} & Out(D̃_{Y}).}

The diagram commutes, since for a given f : BY → BY, both compositions give a map BT̃ → BT̃ over f̃: BỸ → BỸ, and hence they give the same element in Out(D̃_{Y}).

By Proposition 8.10(2), D̃_{X} ⟶^{≅} { \atop D_{X/Z}}, and, chasing through the definitions, BX̃ ⟶^{≃}

B{ \atop X/Z}. By the fundamental group formula, Theorem 8.6, π_{1}(D_{X/Z}) ⟶^{≅} π_{1}(X/Z). (Note that for the proof of Theorem 1.2 we can assume that X/Z is determined by D_{X/Z} making this reference to Theorem 8.6 unnecessary.) Hence applying diagram (2.1) with Y = X/Z and using the aforementioned identifications for Y = X/Z produces the diagram

\xymatrix{Out(BX/Z) \ar[d] \ar@{->>}[r]^{Φ} & Out(D_{X/Z}) \ar[d]^{≅} \\ Out(BX̃) \ar[r]^{Φ} & Out(D̃_{X}).}

Here the right-hand vertical map is an isomorphism by Proposition 8.10(2) and Corollary 8.16, and the top horizontal map is surjective by assumption. Hence Φ : Out(BX̃) → Out(DX̃) is also surjective.

By [45, Thm. 5.4] there is a short exact sequence BA ⟶^{i} BX′ → BX, BX′ = BX̃ × BZ(X)_{1}, where A is a finite p-group and i : BA → BX′ is central. Proposition 8.4(2) shows that X′ has ℤ_{p}-root datum D_{X′} = (W, L_{0}, {ℤ_{p}b_{σ}}) × (1, L^{W} , ∅) and we have the identification D_{X} ≅ D_{X′}/A as in Proposition 8.14.

We are now ready to show that Out(BX) → Out(D_{X}) is surjective, by lifting an arbitrary element α ∈ Out(D_{X}) to Out(BX). By Proposition 8.14, α identifies with an element α′ ∈ Out(D_{X′}) with α′(A) = A. Since Φ : Out(BX̃) → Out(DX̃) is surjective it follows from Proposition 8.13 that Φ : Out(BX′) → Out(D_{X′}) is surjective, so we can find a self-homotopy equivalence φ of BX′ with Φ(φ) = α′. Since α′(A) = A there exists a lift φ′ : BA → BA of φ fitting into a homotopy commutative diagram

\xymatrix{BA \ar[d]_{i} \ar[r]^{φ′} & BA \ar[d]^{i} \\ BX′ \ar[r]^{φ} & BX′.}

Finally, since X ≅ X′/A, Construction 2.2 now gives a self-homotopy equivalence \overline{φ} of BX with the property that Φ(\overline{φ}) = Φ(φ)/A = α′/A = α as desired. □

3. Preliminaries on self-equivalences of non-connected p-compact groups

In this short section we prove a fact about detection of self-equivalences of non-connected p-compact groups on maximal torus normalizers, which we need in the proof of the main theorem, where non-connected groups occur as centralizers of elementary abelian p-groups in connected groups.

Proposition 3.1. Let X be a (not necessarily connected) p-compact group with maximal torus normalizer N and identity component X_{1}. If Φ : Out(BX_{1}) ⟶^{≅} Out(D_{X_{1}}) is an isomorphism, then Φ : Out(BX) → Out(BN) is injective.

Proof. Let j : BN → BX be a normalizer inclusion map, which we can assume is a fibration. Let f : BX → Bπ_{0}(X) = Bπ be the canonical fibration and set q = f ∘ j.

We first argue that we can make the identification

π_{0}(Aut(q)) ⟶^{≅} {φ ∈ Out(BN) | φ(ker(π_{1}(q))) = ker(π_{1}(q))}.

Surjectivity is obvious, so we have to see injectivity, where we first observe that we can pass to a discrete approximation q̆: BN̆ → Bπ, where BN̆ and Bπ are the standard bar construction models. The simplicial maps BN̆ → BN̆ are exactly the group homomorphisms, so any map φ : q̆ → q̆ with BN̆ → BN̆ homotopic to the identity is induced by conjugation by an element in N̆. Hence φ is homotopic to the identity as a map of fibrations, proving the claim.

Since B Aut(f) ⟶^{≃} B Aut(BX), we have the following diagram, with horizontal maps fibrations, where BN_{1} denotes the fiber of BN → Bπ:

[figure: map(Bπ, B Aut(BX_{1}))_{C(f)}; B Aut(BX); B Aut(Bπ); map(Bπ, B Aut(BN_{1}))_{C(q)}; B Aut(q); B Aut(Bπ).]

Here the horizontal fibrations are established in [20] (see also [26, Prop. 11.9]) and the vertical maps are induced by Adams-Mahmud maps, cf. [8, Lem. 4.1], so the diagram is homotopy commutativity by the naturality of these maps.

To establish the proposition it is enough to verify that π_{1}(B Aut(BX)) → π_{1}(B Aut(q)) is injective since we already saw that π_{1}(B Aut(q)) injects into Out(BN). By [4, Thm. B] the map Φ : B Aut(BX_{1}) → B Aut(BN_{1}) factors through the covering space Y of B Aut(BN_{1}) with respect to the subgroup Out(BN_{1}, {(BN_{1})_{σ}}), and B Aut(BX_{1}) → Y has left homotopy inverse. Since Y → B Aut(BN_{1}) is a covering, map map(Bπ, Y) → map(Bπ, B Aut(BN_{1})) is likewise a covering map over each component where it is surjective, and hence induces a monomorphism on π_{1} on all components of map(Bπ, Y). Since B Aut(BX_{1}) → Y has a homotopy retract, map(Bπ, B Aut(BX_{1})) → map(Bπ, B Aut(BN_{1})) also induces an injection on π_{1} for all choices of base-point. Hence the five-lemma and the longexact sequence in homotopy groups applied to the pair of fibration above guarantees that Out(BX) = π_{1}(B Aut(BX)) → π_{1}(B Aut(q)) is injective as wanted. □

Proposition 3.2. Suppose that X is a (not necessarily connected) p-compact group such that Φ : Out(BX_{1}) ⟶^{≅} Out(D_{X_{1}}). Let i : BN_{p} → BX be the inclusion of a p-normalizer of a maximal torus, and let φ : BX → BX be a self-homotopy equivalence. If φ ∘ i is homotopic to i then φ is homotopic to the identity map.

Proof. Let j : BN → BX be the inclusion of a maximal torus normalizer, turned into a fibration. By construction of the Adams-Mahmud map, cf. [8, Lem. 4.1], φ lifts to a map φ′ : BN → BN, making the diagram

\xymatrix{BN \ar[r]^{φ′} \ar[d]^{j} & BN \ar[d]^{j} \\ BX \ar[r]^{φ} & BX}

(strictly) commute, and the space of such lifts is contractible.

We want to see that φ′ is homotopic to the identity, since Proposition 3.1 then implies that φ is homotopic to the identity, as wanted, using the assumption that Φ : Out(BX_{1}) ⟶^{≅} Out(D_{X_{1}}).

Lift i : BN_{p} → BX to a map k : BN_{p} → BN. By [8, Pf. of Lem. 4.1] the space of such lifts is contractible, so since φ ∘ i is homotopic to i, we conclude that φ′ ∘ k is homotopic to k. Replacing k and φ′ by discrete approximations, we get the following diagram which commutes up to homotopy

\xymatrix{ & BN̆_{p} \ar[dr]^{k̆} \ar[dl]_{k̆} \\ BN̆ \ar[rr]^{φ̆′} &  & BN̆ .}

This is a diagram of K(π, 1)’s, so after changing the spaces and maps up to homotopy we can assume that all maps are induced by group homomorphisms, and that the diagram commutes strictly. But now φ̆′ is a group homomorphism which is the identity on im(k̆), and in particular it is the identity on T̆. Hence φ̆′/T̆ : W → W is the identity on the Weyl group W_{1} of X_{1}, since W_{1} acts faithfully on T̆. But W is generated by W_{1} and the image of im(k̆), so we conclude that φ̆′/T̆ : W → W is the identity as well. Hence φ̆′ is in the image of Der(W; T̆) → Aut(N̆), cf. [8, Pf. of Prop. 5.2], and the homotopy class of φ̆′ is the image of an element in H^{1}(W; T̆) under the homomorphism H^{1}(W; T̆) → Out(N̆). Since φ̆′ is the identity on im(k̆), this element restricts trivially to H^{1}(W_{p}; T̆), where W_{p} is a Sylow p-subgroup in W. By a transfer argument the element in H^{1}(W; T̆) is therefore also trivial, and φ̆′ is homotopic to the identity map, as wanted. □

4. First part of the proof of Theorem 1.2: Maps on centralizers

In this section we carry out the first part of the proof of Theorem 1.2, by constructing maps on certain centralizers. We have chosen to be quite explicit about when we replace spaces by homotopy equivalent spaces, since some of these issues become important later on in the proof, where we want to conclude that various constructions really take place in certain over- or undercategories.

Recall that our setup is as follows: Let X and X′ be connected center-free simple pcompact groups with isomorphic ℤ_{p}-root data D_{X} and D_{X′}. We want to prove that BX is homotopy equivalent to BX′, by induction on cohomological dimension, where by [27, Lem. 3.8] the cohomological dimension cd Y of a connected p-compact group Y depends only on D_{Y}. We make the following inductive hypothesis:

(⋆) For all connected p-compact groups Y with cd Y < cd X, Y is determined by its

ℤ_{p}-root datum D_{Y} and Φ : Out(BY) → Out(D_{Y}) is an isomorphism. Let D be a fixed ℤ_{p}-root datum, isomorphic to both D_{X} and D_{X′} and let N = N_{D}

denote the associated normalizer; see Section 8.1. By [4, Thm. 3.1(2)] we can choose maps j : BN → BX and j′ : BN → BX′ making N a maximal torus normalizer in both X and X′ in such a way that the root subgroups BN_{σ} of BN with respect to j : BN → BX and j′ : BN → BX′ agree.

By Theorem 8.6, X and X′ have canonically isomorphic fundamental groups, which both identify with π_{1}(D) via the inclusions j and j′. Applying the second Postnikov section P_{2} to BX and BX′ hence gives us a diagram

\xymatrix{ & BN \ar[dr]^{j′} \ar[dl]_{j} \\ BX \ar[dr] &  & BX′ \ar@{-->}[dl] \\  & B^{2}π_{1}(D)}

where we by changing BN and the maps up to homotopy can assume that all maps are fibrations, and that the diagram commutes strictly. After doing this, we now leave these maps fixed throughout the proof.

Suppose that ν : BV → BX is a rank one elementary abelian p-subgroup of X, and let μ : BV → BT → BN denote the factorization of ν through the maximal torus T, which exists by [25, Prop. 5.6], and is unique up to conjugacy in N by [27, Prop. 3.4]. Set ν′ = j′ ∘ μ for short.

Taking centralizers, these maps produce the following diagram

(4.1)  \xymatrix{ & BC_{N} (μ) \ar[dr]^{j′} \ar[dl]_{j} \\ BC_{X}(ν) &  & BC_{X′}(ν′).}

where we by a slight abuse of notation keep the labeling j and j′. Now the fundamental groups of BC_{X}(ν) and BC_{X′}(ν′) identify via j and j′ with a certain quotient group π of π_{1}(BC_{N} (μ)), explicitly described in [26, Rem. 2.11] and Proposition 8.4(3). Passing to the universal cover of BC_{X}(ν) and BC_{X′}(ν′) and the cover of BC_{N} (μ) determined by the kernel of π_{1}(BC_{N} (μ)) → π produces a diagram

(4.2)  \xymatrix{ & BC_{N} (μ)_{1} \ar[dr]^{j′_{ν′}} \ar[dl]_{j_{ν}} \\ BC_{X}(ν)_{1} &  & BC_{X′}(ν′)_{1}}

where the maps, which are the covers of j and j′, are π-equivariant with respect to the natural free action of π on all three spaces in the diagram.

Note that in general if Y is a space with a map f : Y → BG, with BG the classifying space of a simplicial group G, a specific model for the homotopy fiber Ỹ of f is given by the subspace of Y × EG, consisting of pairs whose images in BG agree. In particular it has a canonical free G-action, via the action on the second coordinate and the projection map Ỹ → Y induces a homotopy equivalence Ỹ/G → Y. We use this model ˜(·) for the homotopy fiber in what follows. Note that if Y has a free H-action, then Ỹ has a free G × H-action.

The spaces in (4.2) all have maps to B^{2}π_{1}(D) making the obvious diagrams commute, so we can take homotopy fibers of these maps by pulling back along the map EBπ_{1}(D) → B^{2}π_{1}(D), as described above. This produces the diagram

(4.3)  \xymatrix{ & B{ \atop C_{N} (μ)_{1}} \ar[dr]^{j̃′_{ν′}} \ar@{-->}[dl]_{j̃_{ν}} \\ B{ \atop C_{X}(ν)_{1}} &  & B{ \atop C_{X′}(ν′)_{1}}.}

Note that by construction K = π × Bπ_{1}(D) acts freely on the spaces in (4.3), and the maps are K-equivariant. By Propositions 8.4(3) and 8.9, B{ \atop C_{X}(ν)_{1}} and B{ \atop C_{X′}(ν′)_{1}} have isomorphic ℤ_{p}-root data and strictly smaller cohomological dimension than X, so the inductive assumption (⋆) guarantees that they are homotopy equivalent. Furthermore by construction j̃_{ν} and j̃′_{ν′} are both maximal torus normalizers and they define the same root subgroups

in B{ \atop C_{N} (μ)_{1}}, since this is true for j and j′. Therefore, by the above and the inductive hypothesis (⋆) there exists a map φ : B{ \atop C_{X}(ν)_{1}} → B{ \atop C_{X′}(ν′)_{1}}, unique up to homotopy, making the above diagram (4.3) homotopy commute. We now want to argue that this map can be chosen to be K-equivariant so that passing to a quotient of diagram (4.3) with φ inserted produces a left-to right map making the diagram (4.1) homotopy commute.

Consider the Adams-Mahmud-like zig-zag

(4.4) Ξ : map(B{ \atop C_{X}(ν)_{1}}, B{ \atop C_{X′}(ν′)_{1}})_{φ} ⟵^{≃} map(j̃_{ν}, j̃′_{ν′})_{φ} ⟶^{≃} map(B{ \atop C_{N} (μ)_{1}}, B{ \atop C_{N} (μ)_{1})_{1}}

where the first map is a homotopy equivalence by [8, Pf. of Lem. 4.1]. That the composite is a homotopy equivalence will follow once we know that the center of C_{X}(ν)_{1} agrees with the center of C_{N} (μ)_{1}, and this follows from Lemma 8.7 applied to the subgroup { \atop C_{X}(ν)_{1}} of X̃, where the assumptions are satisfied since π_{1}(DX̃) = 0 by Proposition 8.9 and Theorem 8.6.

By construction the maps in (4.4) are equivariant with respect to the K-actions. Likewise, since the action on the sources in the mapping spaces is already free, taking homotopy fixedpoints agrees up to homotopy with taking actual fixed-points, so the maps in (4.4) induce homotopy equivalences between the fixed-points. This produces homotopy equivalences

map_{K}(B{ \atop C_{X}(ν)_{1}}, B{ \atop C_{X′}(ν′)_{1}})_{[φ]} ⟵^{≃} map_{K}(j̃_{ν}, j̃′_{ν′})_{[φ]} ⟶^{≃} map_{K}(B{ \atop C_{N} (μ)_{1}}, B{ \atop C_{N} (μ)_{1}})_{[1]},

where the subscript [φ] denotes that we are taking all components of maps non-equivariantly homotopy equivalent to φ. We can therefore pick an equivariant map ψ ∈ map_{K}(B{ \atop C_{X}(ν)_{1}}, B{ \atop C_{X′}(ν′)_{1}})_{[φ]}

corresponding to 1 ∈ map_{K}(B{ \atop C_{N} (μ)_{1}}, B{ \atop C_{N} (μ)_{1}})_{[1]}. Define h̃_{ν} as the composite

h̃_{ν} : B{ \atop C_{X}(ν)} ⟵^{≃} (B{ \atop C_{X}(ν)_{1}})/π ⟶_{≃}^{ψ/π} (B{ \atop C_{X′}(ν′)_{1}})/π ⟶^{≃} B{ \atop C_{X′}(ν′)}

and similarly define h_{ν} : BC_{X}(ν) ⟵^{≃} (B{ \atop C_{X}(ν)_{1}})/K ⟶_{≃}^{ψ/K} (B{ \atop C_{X′}(ν′)_{1}})/K ⟶^{≃} BC_{X′}(ν′).

By construction the maps h̃_{ν} and h_{ν} fit into following homotopy commutative diagram

\xymatrix{ & B{ \atop C_{N} (μ)} \ar@{-->}[dr]^{j̃′_{ν′}} \ar@{-->}[dl]_{j̃_{ν}} \ar[d] \\ B{ \atop C_{X}(ν)} \ar[d] \ar[rr]_{≃}_{h̃_{ν}} & BC_{N} (μ) \ar[dr]^{j′_{ν′}} \ar[dl]_{j_{ν}} & B{ \atop C_{X′}(ν′)} \ar[d] \\ BC_{X}(ν) \ar[rr]_{≃}_{h_{ν}} &  & BC_{X′}(ν′)}

and are in fact uniquely determined up to homotopy by this property by Proposition 3.1 and the inductive assumption (⋆).

Let φ̃_{ν} : B{ \atop C_{X}(ν)} → BX̃′ be the composite of h̃_{ν} with the evaluation B{ \atop C_{X′}(ν′)} → BX̃′ and similarly define φ_{ν} : BC_{X}(ν) → BX′ as the composite of h_{ν} with the evaluation BC_{X′}(ν′) → BX′.

We define φ_{ν} and φ̃_{ν} when ν : BE → BX is an elementary abelian p-subgroup of rank greater that one, by restricting to a rank one subgroup V ⊆ E and using adjointness as follows: As before ν|_{V} factors through T, uniquely as a map to N, and we let μ : BV → BN denote the resulting map to N. Restriction produces a map

φ_{ν,V} : BC_{X}(ν) → BC_{X}(ν|_{V}) ⟶_{≃}^{h_{ν|_{V}}} BC_{X′}(j′μ) → BX′.

Similarly, letting B{ \atop C_{X}(ν)} denote the homotopy fiber of the map BC_{X}(ν) → BX → B^{2}π_{1}(D) as in the rank one case, we define φ̃_{ν,V} as

φ̃_{ν,V} : B{ \atop C_{X}(ν)} → B{ \atop C_{X}(ν|_{V})} h̃_{ν|_{V}} ; ⟶ ; ≃ B{ \atop C_{X′}(j′μ)} → BX̃′.

By construction φ_{ν,V} and φ̃_{ν,V} fit together in that the diagram

\xymatrix{B{ \atop C_{X}(ν)} \ar[r]^{φ̃_{ν,V}} \ar[d] & BX̃′ \ar[d] \\ BC_{X}(ν) \ar[r]^{φ_{ν,V}} & BX′}

commutes up to homotopy, and is in fact a homotopy pull-back square by construction. This concludes the construction of the maps on centralizers which we will use in the next sections to construct our equivalence BX → BX′. We will in particular prove that φ_{ν,V} and φ̃_{ν,V} are independent of the choice of the rank one subgroup V ⊆ E, after which we will drop the subscript V.

5. Second part of the proof of Theorem 1.2: The element in lim^{0}

In this section we prove that the maps φ̃_{ν,V} constructed in the previous section are independent of the choice of rank one subgroup V ⊆ E and give coherent maps into BX̃′. More specifically we prove the following.

Theorem 5.1. Let X and X′ be two connected simple center-free p-compact groups with isomorphic ℤ_{p}-root data, and assume the inductive hypothesis (⋆). Then the maps

φ̃_{ν,V} : B{ \atop C_{X}(ν)} → BX̃′

constructed in Section 4 are independent of the choice of V and together form an element

in lim_{ν∈A(X)}^{0}[B{ \atop C_{X}(ν)}, BX̃′].

Here A(X) is the Quillen category of X with objects the elementary abelian p-subgroups ν : BE → BX of X and morphisms given by conjugation (i.e., the morphisms from (ν : BE → BX) to (ν′ : BE′ → BX) are the linear maps φ : E → E′ such that ν is freely homotopic to ν′ ∘ Bφ).

We need the following proposition, whose proof we postpone to after the rest of the proof of Theorem 5.1.

Proposition 5.2. Let X be a connected simple center-free p-compact group. If ν : BE → BX is a non-toral elementary abelian p-subgroup of rank two, then C_{X}(ν)_{1} is non-trivial or

D_{X} ≅ D_{PU(p)_{p̂}}.

Proof of Theorem 5.1. We divide the proof into two steps. Step 1 verifies the independence of the choice of V, and the shorter Step 2 then uses this to construct the element in lim^{0}.

Step 1: The maps φ̃_{ν,V} and φ_{ν,V} are independent of the choice of rank one subgroup V: We divide this step into three substeps a–c. Step 1a assumes ν toral, Step 1b assumes ν rank two non-toral, and finally Step 1c considers the general case. Step 1a: Assume ν : BE → BX is a toral elementary abelian p-subgroup. By assumption ν factors through BT to give a map μ : BE → BN, unique up to conjugation in N, and as in the rank one case we let ν′ = j′μ. We want to say that the map φ̃_{ν,V} does not depend

on V, basically since it is a map suitably under B{ \atop C_{N} (μ)}, and hence uniquely determined, independently of V. This will follow by adjointness, analogously to [8, Pf. of Thm. 2.2], although a bit of care has to be taken, since we have to verify that this happens over B^{2}π_{1}(D) in order to be able to pass to the cover ˜(·), as we now explain.

Recall that by construction the map h_{ν|_{V}} is the bottom left-to-right composite in the following diagram (5.1)

\xymatrix{ &  & BC_{N} (μ|_{V}) \ar[ddll]_{j_{ν|_{V}}} \ar[ddrr]^{j′_{ν′|_{V}}} \\  &  & (B{ \atop C_{N} (μ|_{V})_{1}})/K \ar[u]_{≃} \ar@{-->}[dl] \ar@{-->}[dr] \\ BC_{X}(ν|_{V}) & (B{ \atop C_{X}(ν|_{V})_{1}})/K \ar[l]_{≃} \ar[rr]_{≃}^{ψ/K} &  & (B{ \atop C_{X′}(ν′|_{V})_{1}})/K \ar[r]^{≃} & BC_{X′}(ν′|_{V})}

where we notice that all subdiagrams commute up to homotopy over BK.

Since BE maps into these spaces via ν and μ, adjointness produces the following diagram (5.2)

\xymatrix{ &  & BC_{N} (μ) \ar[ddll]_{j_{ν}} \ar[ddrr]^{j′_{ν′}} \\  &  & BC_{C_{N} (μ|_{V} )}(μ) \ar[u]_{≃} \ar[dl] \ar@{-->}[dr] \\ BC_{X}(ν) & BC_{C_{X}(ν|_{V} )}(ν) \ar[l]_{≃} & Z \ar[l]_{≃} \ar[r]_{≃}^{C(ψ/K)} & BC_{C_{X′}(ν′|_{V} )}(ν′) \ar[r]^{≃} & BC_{X′}(ν′)}

where Z = map(BE, (B{ \atop C_{X}(ν|_{V})_{1}})/K)_{ν} and C(ψ/K) is the map induced by ψ/K on mapping spaces.

Since the diagram (5.2) homotopy commutes as a diagram over B^{2}π_{1}(D) we get a homotopy commutative diagram by passing to homotopy fibers: (5.3)

\xymatrix{ &  & B{ \atop C_{N} (μ)} \ar[ddll]_{j̃_{ν}} \ar[ddrr]^{j̃′_{ν′}} \\  &  & B{ \atop C_{C_{N} (μ|_{V} )}(μ)} \ar[u]_{≃} \ar@{-->}[dl] \ar@{-->}[dr] \\ B{ \atop C_{X}(ν)} & B{ \atop C_{C_{X}(ν|_{V} )}(ν)} \ar[l]_{≃} & Z̃ \ar[l]_{≃} \ar[r]_{≃}^{{ \atop C(ψ/K)}} & B{ \atop C_{C_{X′}(ν′|_{V} )}(ν′)} \ar[r]^{≃} & B{ \atop C_{X′}(ν′)}.}

Denote the bottom left-to-right homotopy equivalence in (5.3) by C_{ν}(h̃_{ν|_{V}}), justified by the fact that by construction the following diagram homotopy commutes

(5.4)  \xymatrix{B{ \atop C_{X}(ν)} \ar[r]^{C_{ν}(h̃_{ν|_{V}} )}_{≃} \ar[d] & B{ \atop C_{X′}(ν′)} \ar[d] \ar[dr] \\ B{ \atop C_{X}(ν|_{V})} \ar[r]^{h̃_{ν|_{V}}}_{≃} & B{ \atop C_{X′}(ν′|_{V})} \ar[r] & BX̃′.}

Diagram (5.3), together with the inductive assumption (⋆) and Proposition 3.1 shows that the homotopy class of C_{ν}(h̃_{ν|_{V}}) does not depend on V. Hence by diagram (5.4) the same is true for φ̃_{ν,V} which is what we wanted. (The key point in the above argument is that we can choose μ once and for all, such that μ|_{V} is a factorization of ν|_{V} through BT for every V ⊆ E.)

Note that the construction of C_{ν}(h̃_{ν|_{V}}) in (5.3) does not depend on ν being toral, as long

as ν′ in that case is defined as BE ⟶^{ν} BC_{X}(ν) ⟶^{φ_{ν,V}} BX′ (instead of j′μ), which makes sense in this more general setting—the top part of diagram (5.3) is only needed to conclude the independence of V. In Step 1b below we will also use the notation C_{ν}(h̃_{ν|_{V}}) for non-toral ν.

Step 1b: Assume ν : BE → BX is a rank two non-toral elementary abelian p-subgroup. By Proposition 5.2 either C_{X}(ν)_{1} is non-trivial or D_{X} ≅ D_{PU(p)_{p̂}}.

Assume first that D_{X} ≅ D_{PU(p)_{p̂}}. Since uniqueness for this group is well known, both for p odd and p = 2 by [21], [43], [11] and [8], the statement of course follows for this reason. But one can also argue directly, using a slight modification of the proof of [8, Lem. 3.2] which we quickly sketch: For α ∈ W_{X}(ν) we have the following diagram

[figure: B{ \atop C_{X}(ν)}; B{ \atop C_{X}(ν|_{V})}; h̃_{ν|_{V}}; ≃; B{ \atop C_{X′}(j′μ)}; BX̃′; B{ \atop C_{X}(ν)}; B{ \atop C_{X}(ν|_{α(V)})}; { \atop h_{ν|_{α(V )}}}; ≃; B{ \atop C_{X′}(j′μ ∘ (α^{−1}|_{α(V)}))}; BX̃′.]

Here all the non-identity vertical maps are given on the level of mapping spaces (i.e., without the tilde) by f ↦ f ∘α^{−1}, which induces a map on the indicated spaces by taking homotopy fibers of the map to B^{2}π_{1}(D). The left-hand and right-hand squares obviously commute and the middle square commutes up to homotopy by our inductive assumption (⋆) and Proposition 3.1. We thus conclude that { \atop φ_{ν,α(V)}} ∘ { \atop BC_{X}(α^{−1})} ≃ φ̃_{ν,V} for all α ∈ W_{X}(ν). Now, since D ≅ D_{PU(p)_{p̂}} we have π_{1}(D) ≅ ℤ/p, and because E is non-toral, we see that

BC_{X}(ν) ≃ BE and B{ \atop C_{X}(ν)} ≃ BP, where P is the extra-special group p_{+}^{1+2} of order p^{3} and exponent p if p is odd and Q_{8} if p = 2. Now [8, Prop. 3.1] shows that W_{X}(ν) contains SL(E), and the same is true for X′. The proof of [8, Prop. 3.1] furthermore shows that φ̃_{ν,V} ∘ { \atop BC_{X}(α)} ≃ φ̃_{ν,V} for α ∈ SL(E). (Apply the argument there to the p-group P instead of E.) Since SL(E) acts transitively on the rank one subgroups of E, combining the above gives that φ_{ν,V′} ≃ φ̃_{ν,V} for any rank one subgroup V′ ⊆ E as desired.

We can therefore assume that C_{X}(ν)_{1} is non-trivial, and the proof in this case is an adaptation of [8, Pf. of Lem. 3.3] to our new setting: Choose a rank one elementary abelian p-subgroup η : BU = Bℤ/p → BC_{X}(ν)_{1} in the center of a p-normalizer of a maximal torus in C_{X}(ν). Let η × ν : BU × BE → BX be the map defined by adjointness, and for any rank one subgroup V of E, consider the map η × ν|_{V} : BU × BV → BX obtained by restriction. By construction η × ν|_{V} is the adjoint of the composite BU ⟶^{η} BC_{X}(ν)_{1} ⟶^{res} BC_{X}(ν|_{V})_{1}, so η × ν|_{V} : BU × BV → BX factors through a maximal torus in X by [25, Prop. 5.6]. It is furthermore straightforward to check that η × ν is a monomorphism (compare [8, Pf. of Lem. 3.3]).

Now consider the following diagram

\xymatrix{ & BC_{X}(ν|_{V}) \ar[dr]^{φ_{ν|_{V}}} \\ BU × BE \ar[r] \ar[ur] \ar[dr] & BC_{X}(η × ν|_{V}) \ar[u] \ar[d] & BX′ \\  & BC_{X}(η) \ar[ur]_{φ_{η}}}

Here the left-hand side of the diagram is constructed by taking adjoints of η × ν and hence it commutes. The right-hand side homotopy commutes by Step 1a (using the inductive assumption (⋆)), since η × ν|_{V} is toral of rank two. We can hence without ambiguity define (η × ν)′ as either the top left-to-right composite (for some rank one subgroup V ⊆ E) or the bottom left-to-right composite. Denote by ν′ the restriction of (η × ν)′ to BE.

By construction of the map C_{η×ν}(h̃_{ν|_{V}}) as the bottom composite in (5.3), the diagram

(5.5)  \xymatrix{B{ \atop C_{X}(η × ν)} \ar[r]^{C_{η×ν}(h̃_{ν|_{V}} )}_{≃} \ar[d] & B{ \atop C_{X′}((η × ν)′)} \ar[d] \\ B{ \atop C_{X}(ν)} \ar[r]^{C_{ν}(h̃_{ν|_{V}} )}_{≃} & B{ \atop C_{X′}(ν′)}}

commutes. Furthermore, since η × ν|_{V} is toral, diagram (5.3), applied with μ equal to a factorization of η × ν|_{V} through BT, shows that the top horizontal map in (5.5) agrees with C_{η×ν}(h̃_{η}), and in particular it is independent of V, again using Proposition 3.1 and our inductive assumption (⋆).

We claim that this forces the same to be true for the bottom horizontal map in (5.5): By our choice of η, the centralizer C_{X}(η × ν) contains a p-normalizer of a maximal torus in C_{X}(ν), and hence { \atop C_{X}(η × ν)} contains a p-normalizer of a maximal torus in { \atop C_{X}(ν)}. Proposition 3.2 and our inductive assumption (⋆) therefore shows that the bottom map in (5.5)

is independent of V, so φ̃_{ν,V} : B{ \atop C_{X}(ν)} ⟶^{C_{ν}(h̃_{ν|_{V}} )} B{ \atop C_{X′}(ν′)} → BX̃′ is also independent of V as wanted.

Step 1c: Assume ν : BE → BX is an elementary abelian p-subgroup of rank ≥ 3. The fact that φ̃_{ν,V} is independent of V when E has rank two implies the statement in general: Let ν : BE → BX be an elementary abelian p-subgroup of rank at least three, and suppose that V_{1} ≠ V_{2} are two rank one subgroups of E. Setting U = V_{1} ⊕ V_{2} we get the following diagram

\xymatrix{ & B{ \atop C_{X}(ν|_{V_{1}})} \ar[dr]^{{ \atop φ_{ν|_{V_{1}}}}} \\ B{ \atop C_{X}(ν)} \ar@{-->}[ur] \ar[r] \ar[dr] & B{ \atop C_{X}(ν|_{U})} \ar[u] \ar[d] & BX̃′ \\  & B{ \atop C_{X}(ν|_{V_{2}})} \ar@{-->}[ur]_{{ \atop φ_{ν|_{V_{2}}}}}}

The left-hand side of this diagram is constructed by adjointness and hence commutes, and the right-hand side of the diagram commutes up to homotopy by Steps 1a and 1b. Thus the top left-to-right composite { \atop φ_{ν,V_{1}}} is homotopic to the bottom left-to-right composite { \atop φ_{ν,V_{2}}}, i.e., the map φ̃_{ν,V} is independent of the choice of rank one subgroup V as claimed.

Step 2: An element in lim^{0}. With the above preparations in place it is easy to see, as in [8, Pf. of Thm. 2.2], that the maps φ̃_{ν} : B{ \atop C_{X}(ν)} → BX̃′ fit together to form an element in

lim _{ν∈A(X)}^{0}[B{ \atop C_{X}(ν)}, BX̃′].

In order not to cheat the reader of the finale, we repeat the short argument from [8, Pf. of Thm. 2.2]: For any morphism ρ : (ν : BE → BX) → (ξ : BF → BX) in A(X) we need to verify that

\xymatrix{B{ \atop C_{X}(ξ)} \ar[rr]^{B{ \atop C_{X}(ρ)}} \ar[dr]_{φ̃_{ξ}} &  & B{ \atop C_{X}(ν)} \ar[dl]^{φ̃_{ν}} \\  & BX̃′}

commutes up to homotopy. If F has rank one, then ρ is an isomorphism, and we let μ : BF → BT → BN be a factorization of ξ through BT. In this case the claim follows since

(5.6)  \xymatrix{B{ \atop C_{X}(ξ)} \ar[r]_{≃}^{h̃_{ξ}} \ar[d]^{≃}_{B{ \atop C_{X}(ρ)}} & B{ \atop C_{X′}(j′μ)} \ar[d]^{≃}_{B{ \atop C_{X′}(ρ)}} \ar[dr] \\ B{ \atop C_{X}(ξρ)} \ar[r]^{h̃_{ξρ}}_{≃} & B{ \atop C_{X′}(j′μρ)} \ar[r] & BX̃′}

commutes up to homotopy, since we can view the diagram as taking place under B{ \atop C_{N} (μ)} ⟶^{≃} B{ \atop C_{N} (μρ)}, up to homotopy, using the inductive assumption (⋆) and Proposition 3.1 as in Step 1a. The case where F has arbitrary rank follows from the independence of the choice of rank one subgroup V, established in Step 1, together with the rank one case: For a rank one subgroup V of E set V′ = ρ(V) and consider the diagram

\xymatrix{B{ \atop C_{X}(ξ)} \ar[r] \ar[d]_{B{ \atop C_{X}(ρ)}} & B{ \atop C_{X}(ξ|_{V′})} \ar[d]_{B{ \atop C_{X}(ρ|_{V} )}} \ar[dr] \\ B{ \atop C_{X}(ν)} \ar[r] & B{ \atop C_{X}(ν|_{V})} \ar[r] & BX̃′}

The left-hand side commutes by construction and the right-hand side commutes since the diagram (5.6) commutes, proving the claim.

This constructs an element [ϑ] ∈ lim_{ν∈A(X)}^{0}[B{ \atop C_{X}(ν)}, BX̃′] as wanted. □

We now give the proof of Proposition 5.2, used in the proof of Theorem 5.1, which we postponed. The proof uses case-by-case arguments on the level of ℤ_{p}-root data.

Proof of Proposition 5.2. Assume first that (W_{X}, L_{X}) is an exotic ℤ_{p}-reflection group. We claim that the centralizer of any rank one elementary abelian p-subgroup of X is connected, which in particular implies that there can be no rank two non-toral elementary abelian p-subgroups: For p > 2 this follows from [8, Thms. 11.1, 12.2(2) and 7.1] combined with Dwyer-Wilkerson’s formula for the Weyl group of a centralizer [26, Thm. 7.6]. For p = 2 we have D_{X} ≅ D_{DI(4)} and hence W_{X} ≅ ℤ/2×GL_{3}(𝔽_{2}) where the central ℤ/2 factor acts by −1 on L_{X} and GL_{3}(𝔽_{2}) acts via the natural representation on L_{X} ⊗ 𝔽_{2}, cf. [8, Pf of Thm. 11.1] or [28, Rem. 7.2]. In particular it follows directly (cf. [28, Pf. of Prop. 9.12, DI(4) case]) that X contains a unique elementary abelian 2-subgroup of rank one up to conjugation and that the centralizer of this subgroup is connected.

By the classification of ℤ_{p}-root data, Theorem 8.1(1), we may thus assume that D_{X} is of the form D_{G_{p̂}} for some simple compact connected Lie group G, and since X is center-free we may assume that so is G. If π_{1}(G) has no p-torsion, [59, Thm. 2.27] implies that the centralizer in G of any element of order p is connected. By the formula for the Weyl group of a centralizer [26, Thm. 7.6] (cf. Proposition 8.4(3)), it then follows that C_{X}(η) is connected for any rank one elementary abelian p-subgroup η : Bℤ/p → BX of X, and hence X does not have any rank two non-toral elementary abelian p-subgroups.

We can thus furthermore assume that π_{1}(G) has p-torsion, which implies that D_{X} ≅ D_{G_{p̂}} for one of the following G: G = PU(n) for p|n; G = SO(2n+1), n ≥ 2 for p = 2; G = PSp(n), n ≥ 3 for p = 2; G = PSO(2n), n ≥ 4 for p = 2; G = P E_{6} for p = 3 and G = P E_{7} for p = 2. We want to see that if η : Bℤ/p → BX has rank one and D_{X} ≅ D̸_{PU(p)_{p̂}}, then C_{X}(η) is not a p-compact toral group, since then [23, Cor. 1.4] implies that for any elementary abelian p-subgroup ν of rank two, C_{X}(ν)_{1} is non-trivial.

By [26, Thm. 7.6] it is enough to see this for the corresponding Lie group G. So, let V ⊆ G be a rank one elementary abelian p-subgroup of G. If C_{G}(V)_{1} is a torus then W_{C_{G}(V)_{1}} = 1 and hence W_{C_{G}(V)} ≅ π_{0}(C_{G}(V)). By [9, §5, Ex. 3(b)] or [58, Thm. 9.1(a)] π_{0}(C_{G}(V)) is isomorphic to a subgroup of π_{1}(G), so |W_{C_{G}(V)}| divides |π_{1}(G)|. In particular, if x is a generator of V then the number of elements in a maximal torus which are conjugate to x in G is at least |W_{G}|/|π_{1}(G)| (since two elements in a maximal torus are conjugate in G if and only if they are conjugate by a Weyl group element). In particular

p^{n} − 1 ≥ \frac{|W_{G}|}{|π_{1}(G)|},

where n is the rank of G. A direct case-by-case check of the above cases shows that this inequality can only hold when G = PU(n), p|n. In this case, a generator of V must have the form x = [diag(λ_{1}, …, λ_{n})]. For n > p, some of the λ’s must agree, so C_{G}(V)_{1} is not a torus in this case. This proves the claim. □

6. Third and final part of the proof of Theorem 1.2: Rigidification

In this section we finish the proof of Theorem 1.2 by showing that our element in lim^{0} from Theorem 5.1 rigidifies to produce a homotopy equivalence BX → BX′. We first need a lemma:

Lemma 6.1. Suppose that we have a homotopy pull-back square of p-compact groups

\xymatrix{BX′ \ar[d]^{g′} \ar[r]^{f′} & BY′ \ar[d]^{g} \\ BX \ar[r]^{f} & BY}

where f : BX → BY is a centric monomorphism (i.e., map(BX, BX)_{1} ⟶^{≃} map(BX, BY)_{f} and Y/X is 𝔽_{p}-finite) and g : BY′ → BY is an epimorphism (i.e., Y/Y′ is the classifying space of a p-compact group). Then f′ : BX′ → BY′ is a centric monomorphism, and g′ is an epimorphism.

Proof. It is clear from the pull-back square that f′ is a monomorphism and that g′ is an epimorphism. To see that f′ is centric, observe that we have a map of fibrations

[figure: (Y/Y′)^{hX′}; map(BX′, BX′)_{[g′]}; map(BX′, BX)_{g′}; (Y/Y′)^{hX′}; map(BX′, BY′)_{[f∘g′]}; map(BX′, BY)_{f∘g′}]

where e.g., the subscript [g′] denotes the components of the mapping space mapping to the component of g′. Since the wanted map map(BX′, BX′)_{1} → map(BX′, BY′)_{f′} is the restriction of the map of total spaces to a component, it is hence enough to see that the map between the base spaces is an equivalence, which follow since we have equivalences

\xymatrix{map(BX, BX)_{1} \ar[r]^{≃} \ar[d]^{≃} & map(BX′, BX)_{g′} \ar[d] \\ map(BX, BY)_{f} \ar[r]^{≃} & map(BX′, BY)_{f∘g′}.}

Here the vertical equivalence follows from the centricity of f and the horizontal equivalences follows from [18, Prop. 3.5] combined with [26, Prop. 10.1] (in [18, Prop. 3.5] taking E = BX′, B = BX, and X = BX and BY respectively). □

Proof of Theorem 1.2. By the results of Section 2 (Propositions 2.1 and 2.4) we are reduced to the case where we consider two simple, center-free p-compact groups BX and BX′ with the same root datum D. Furthermore, in Theorem 5.1 of the previous section we constructed an element [ϑ] ∈ lim_{ν∈A(X)}^{0}[B{ \atop C_{X}(ν)}, BX̃′]. We want to use this element to construct the map from BX to BX′, and show that it is an equivalence. By the classification of ℤ_{p}-root data, Theorem 8.1(1), D is either exotic or D ≅ D_{G_{p̂}} for a compact connected Lie group G and we handle these two cases separately.

Suppose that D is exotic, and notice that in this case π_{1}(D) = 0 by Theorem 8.1(2). If p is odd we are exactly in the situation covered by rather easy arguments in [8] (see [8, Pf. of Thm. 2.2] and [8, Prop. 9.5]). For p = 2 uniqueness of B DI(4), as well as the statements about self-maps, are already well known by the work of Notbohm [53], but we nevertheless quickly remark how a proof also falls out of the current setup, noticing that the arguments from [8] from this point on, in the special case of DI(4), carries over verbatim: We have D ≅ D_{DI(4)} and since π_{1}(D) = 0, the functor ν ↦ π_{i}(map(B{ \atop C_{X}(ν)}, BX̃′)_{ν}) identifies with ν ↦ π_{i}(BZ(C_{X}(ν))), as explained in detail in [8, Pf. of Thm. 2.2]. Since we are considering X = DI(4) we know by Dwyer-Wilkerson [24, Prop. 8.1] that lim_{A(X)}^{j} π_{i}(BZ(C_{X}(ν))) = 0 for all i, j ≥ 0. (The proof of this is a Mackey functor argument, and relies on the regular structure of the Quillen category of DI(4), due to the fact that its classifying space, like the exotic p-compact groups for p odd, has polynomial 𝔽_{p}-cohomology ring.) The centralizer decomposition theorem [26, Thm. 8.1] now produces a map BX → BX′, which by standard arguments given in [8, Pf. of Thm. 2.2] is seen to be an equivalence. The statement about self-maps also follows as in [8, Pf. of Thm. 2.2], using Out(BN , {BN_{σ}}) instead of Out(BN).

Now, suppose that D ≅ D_{G_{p̂}}, for a simple center-free compact Lie group G, with universal cover G̃. Let O_{p}^{r}(G̃) be the full subcategory of the orbit category with objects the G̃-sets G̃/P̃ with P̃ a p-radical subgroup (i.e, P̃ is an extension of a torus by a finite p-group, such that NG̃(P̃)/P̃ is finite and contains no non-trivial normal p-subgroups).

For the p-radical homology decomposition [32, Thm. 4] of the compact Lie group G̃, one considers the functor F : O_{p}^{r}(G̃) → Spaces given by G̃/P̃ ↦ EG̃ ×G̃ G̃/P̃, where Spaces denotes the category of topological spaces. Viewed as a functor to the homotopy category of spaces, Ho(Spaces), this functor is isomorphic to the functor F′ : O_{p}^{r}(G̃) → Ho(Spaces)

given on objects by G̃/P̃ ↦ BP̃ and on morphisms by sending the G̃-set map G̃/P̃ ⟶^{f} G̃/Q̃ to the map c_{g^{−1}} : BP̃ → BQ̃, where gQ̃ = f(eP̃), via the canonical equivalences BP̃ = (EP̃)/P̃ ⟶^{≃} EG̃ ×G̃ G̃/P̃. (Note that F′ is not well-defined as a functor to Spaces, since the element g is just an arbitrary coset representative for the coset gQ̃.) We can hence in what follows replace EG̃ ×G̃ G̃/P̃ by BP̃ in this way, whenever we are working in the homotopy category.

Since P̃ and Q̃ are p-radical, the same is true for their images P and Q in G, and C_{G}(P) = Z(P) and likewise for Q (see [32, Prop. 1.6(i) and Lem. 1.5(ii)]). Hence there is a well defined induced morphism c_{g} : _{p}Z(Q) → _{p}Z(P) as well as a well-defined (free)

homotopy class of maps c_{g^{−1}} : B{ \atop C_{G}(_{p}Z(P))} → B{ \atop C_{G}(_{p}Z(Q))}. Consider the diagram

\xymatrix{BP̃_{p̂} \ar[r] \ar[d]^{c_{g^{−1}}} & (B{ \atop C_{G}(_{p}Z(P))})_{p̂} \ar[r]^{≃} \ar[d]^{c_{g^{−1}}} & B{ \atop C_{G_{p̂}}(i_{(_{p}Z(P))})} \ar[dr]^{{ \atop φ_{i_{(pZ(P))}}}} \ar[d]_{BC_{Gp̂}(c_{g})} \\ BQ̃_{p̂} \ar[r] & (B{ \atop C_{G}(_{p}Z(Q))})_{p̂} \ar[r]^{≃} & B{ \atop C_{G_{p̂}}(i_{(_{p}Z(Q))})} \ar[r]_{{ \atop φ_{i_{(pZ(Q))}}}} & BX̃′}

where i_{V} : BV → BG_{p̂} denotes the map induced by the inclusion of a subgroup V ⊆ G, and where the horizontal maps in the middle square are given by lifting the standard homotopy equivalence given by adjointness to the covers. The first two squares are homotopy commutative by construction, and the right-hand triangle commutes since [ϑ] ∈ lim_{ν∈A(X)}^{0}[B{ \atop C_{X}(ν)}, BX̃′]. Hence this diagram produces an element [ζ] ∈ lim_{G̃/P̃∈O_{p}^{r}(G̃)}^{0}[BP̃_{p̂}, BX̃′].

Denote the composition BP̃_{p̂} → B{ \atop C_{G}(_{p}Z(P))}_{p̂} → B{ \atop C_{G_{p̂}}(i_{_{p}Z(P)})} ⟶^{φ_{i_{(pZ(P))}}} BX̃′ by ψ̃_{G̃/P̃}

(i.e., the coordinate of [ζ] corresponding to G̃/P̃), and let ψ_{G/P} : BP_{ p̂} → BX′ be the map constructed analogously, using φ_{i_{(pZ(P))}} instead. We want to lift [ζ] to a map

hocolim_{G̃/P̃∈O_{p}^{r}(G̃)}(EG̃ ×G̃ G̃/P̃)_{p̂} → BX̃′.

By [10, Prop. XII.4.1 and XI.7.1] (see also [66, Prop. 3] or [34, Prop. 1.4]) the obstructions to doing this lie in

lim _{G̃/P̃∈O_{p}^{r}(G̃)}^{i+1}π_{i}(map(BP̃_{p̂}, BX̃′)ψ̃_{G̃/P̃}), i ≥ 1.

By construction, ψ̃_{G̃/P̃} and ψ_{G/P} fit into a homotopy pull-back square

\xymatrix{BP̃_{p̂} \ar[r]^{ψ̃_{G̃/P̃}} \ar[d] & BX̃′ \ar[d] \\ BP_{ p̂} \ar[r]^{ψ_{G/P}} & BX′.}

By [12, Lem. 4.9(1)] the map ψ_{G/P} is centric, so Lemma 6.1 implies that ψ̃_{G̃/P̃} is centric

as well. Hence by centricity and naturality, the functor G̃/P̃ ↦ π_{i}(map(BP̃_{p̂}, BX̃′)ψ̃_{G̃/P̃})

identifies with the functor G̃/P̃ ↦ π_{i−1}(Z(P̃)_{p̂}). Since G̃ is simple and simply connected it now follows from the fundamental calculations of Jackowski-McClure-Oliver [32, Thm. 4.1] that

lim _{O_{p}^{r}(G̃)}^{i+1}π_{i−1}(Z(P̃)_{p̂}) = 0, for i ≥ 1.

Hence by the homology decomposition theorem [32, Thm. 4] we get a map

BG̃_{p̂} ⟵^{≃} (hocolim_{G̃/P̃∈O_{p}^{r}(G̃)}(EG̃ ×G̃ G̃/P̃)_{p̂})_{p̂} → BX̃′

which by construction is a map under BÑ_{p}, the p-normalizer of a maximal torus in BG̃_{p̂}. Dividing out by Z(D) as explained in Construction 2.2 produces the homotopy commutative diagram

\xymatrix{ & BN_{p} \ar[dr] \ar[dl] \\ BG_{p̂} \ar[rr] &  & BX′.}

It is now a short argument, given in detail in [8, Pf. of Thm. 2.2], to see that BX = BG_{p̂} → BX′ is a homotopy equivalence as wanted.

We want to show that Out(BX) → Out(D_{X}) is an isomorphism: To see surjectivity, note that if α ∈ Out(D_{X}), then by [4, Thm. B] and [8, Prop. 5.1], α corresponds to a unique map α′ ∈ Out(BN , {BN_{σ}}). Hence if j : BN → BX is a maximal torus normalizer, then repeating the above argument with respect to the two maps j : BN → BX and j ∘ α′ : BN → BX gives a map BX → BX realizing α ∈ Out(D_{X}). Finally we show injectivity, essentially repeating the argument of Jackowski-McClure-Oliver, cf. [32, Pf. of Thm. 4.2]: We are assuming that BX ≃ BG_{p̂}, for some compact connected center-free Lie group G. As in the proof of Proposition 2.4(3), we have the following commutative diagram

\xymatrix{Out(BG_{p̂}) \ar[d] \ar[r] & Out(D_{G_{p̂}}) \ar[d] \\ Out(BG̃_{p̂}) \ar[r] & Out(D_{G̃_{p̂}}).}

(Compare diagram (2.1), and note that we use that D̃_{G_{p̂}} = DG̃_{p̂} for the right-hand vertical map, which uses Theorem 8.6 for compact Lie groups.) The left-hand vertical map in the above diagram is injective since factoring out by the center (via the quotient construction recalled in Construction 2.2) provides a left inverse (in fact an actual inverse, though we do not need this here). So we just have to see that Out(BG̃_{p̂}) → Out(D_{G̃_{p̂}}) is injective. By [4,

Thm. B], this map factors through Out(BÑ, {BÑ_{σ}}) and Out(BÑ, {BÑ_{σ}}) → Out(D_{G̃_{p̂}})

is an isomorphism, where Ñ denotes the maximal torus normalizer in G̃_{p̂}. By the homology decomposition theorem [32, Thm. 4] and obstruction theory [32, Thm. 3.9] (cf. [66, Prop. 3]), injectivity of Out(BG̃_{p̂}) → Out(BÑ) follows from Jackowski-McClure-Oliver’s calculation of higher limits [32, Thm. 4.1]

lim _{G̃/P̃∈O_{p}^{r}(G̃)}^{i}π_{i−1}(Z(P̃)_{p̂}) = 0, i ≥ 1.

We conclude that Out(BG_{p̂}) → Out(D_{G_{p̂}}) is also injective as claimed.

Finally, the last statement in Theorem 1.2 about the homotopy type of B Aut(BX) follows by combining [4] with what we have proved so far: The Adams-Mahmud map factors as Φ : B Aut(BX) → B Aut(BN , {BN_{σ}}) → B Aut(BN), where B Aut(BN , {BN_{σ}}) is the covering of B Aut(BN) with respect to the subgroup Out(BN , {BN_{σ}}) of the fundamental group. Furthermore [4, §5] explains how killing elements in π_{2}(B Aut(BN , {BN_{σ}})) constructs a space denoted B aut(D_{X}), whose universal cover is B^{2}Z(D_{X}), where BZ(D_{X}) = (BZ̆(D_{X}))_{p̂}. It it furthermore shown there that the fibration B aut(D_{X}) → B Out(D_{X}) is split, i.e., B aut(D_{X}) has the homotopy type of (B^{2}Z(D_{X}))_{h Out(D_{X})}. Composition gives a map B Aut(BX) → B aut(D_{X}), which is an isomorphism on π_{i} for i > 1 by construction, and an isomorphism on π_{1} by what we have shown in the first part of the theorem.

This concludes the proof of the main Theorem 1.2. □

Remark 6.2 (The fundamental group of a p-compact group). In [8], with Møller and Viruel, we established the fundamental group formula for p-compact groups, Theorem 8.6, for p odd, as a consequence of the classification. In this remark we sketch how one by expanding the proof of Theorem 1.2 somewhat can avoid the reliance on Theorem 8.6, hence proving Theorem 8.6 also for p = 2 this way—this was the strategy which we had originally envisioned before Dwyer-Wilkerson [29] provided an alternative direct proof of Theorem 8.6 as we were writing this paper.

The fundamental group formula was not used in the reduction to simple center-free groups, except for a reference in the proof of Proposition 2.4(3), where it, for the purpose of inductively proving Theorem 1.2, was only needed in the well-known case of compact Lie groups. Hence we can assume that X and X′ are simple center-free p-compact groups with the same ℤ_{p}-root datum D, and that we know the fundamental group formula for X and Theorem 1.2 for connected p-compact groups of lower cohomological dimension than X. In the last section we constructed an element [ϑ] ∈ lim_{ν∈A(X)}^{0}[B{ \atop C_{X}(ν)}, BX̃′]. By not passing to a universal cover, and hence not using the fundamental group formula, a simplified version of the same argument gives an element [ϑ̄] ∈ lim_{ν∈A(X)}^{0}[BC_{X}(ν), BX′]. The only non-obvious change is that since the center formula in Lemma 8.7 does not hold in the presence of direct factors isomorphic to D_{SO(2n+1)_{ˆ2}} in the ℤ_{2}-root datum, one has to take the target of Ξ in (4.4) to be BZ(D_{C_{X}(ν)_{1}}) (obtained as a quotient of BZ(C_{N} (μ)_{1}); cf. [4, Lem. 5.1]) to obtain a homotopy equivalence. With this change the rest of Section 4 proceeds as before, but ignoring everything on the level of covers, and one constructs a map like before without any choices. Section 5 has to be modified in the following way: Instead of having maps being under the maximal torus normalizer (which we now do not a priori know), we utilize instead that potentially different maps agree in

M = (map(Bπ, B aut(D)))_{h Aut(Bπ)},

where B aut(D) = (B^{2}Z(D))_{h Out(D)}, which means that they are homotopy equivalent, by the description of self-equivalences of non-connected groups, cf., Theorem 1.6, which we know by induction. From the element in lim^{0} one can easily get an isomorphism between fundamental groups: Consider the diagram

\xymatrix{ & H_{∗}^{ℤ_{p}}(BN_{p}) \ar@{->>}[dl]_{j} \ar[d] \ar@{->>}[dr]^{j′} \\ H_{∗}^{ℤ_{p}}(BX) & colim_{ν∈A(X)} H_{∗}^{ℤ_{p}}(BC_{X}(ν)) \ar[l]_{≅} \ar[r]^{[ϑ̄]} & H_{∗}^{ℤ_{p}}(BX′)}

where the vertical map is given by choosing a central rank one subgroup ρ of the pnormalizer N_{p} and considering the corresponding inclusion N_{p} → C_{X}(j∘ρ). (Here H_{n}^{ℤ_{p}}(Y) = lim H_{n}(Y; ℤ/p^{n}).) Note that the maps j and j′ are surjective by a transfer argument, and that the indicated isomorphism follows by the centralizer homology decomposition theorem [26, Thm. 8.1]. This shows that the kernel of j is contained in the kernel of j′. However, since we could reverse the role of X and X′ in all the previous arguments (we have not used any special model for X), we conclude by symmetry that the kernel of j equals the kernel of j′, and in particular the kernels of the maps j : π_{1}(D) → π_{1}(X) and j′ : π_{1}(D) → π_{1}(X′) agree, and they are surjective by Proposition 8.5. If D is exotic then π_{1}(D) = 0 by Theorem 8.1(2), so there is nothing to prove. If D is not exotic, then by Theorem 8.1(1), D = D_{G} ⊗_{ℤ} ℤ_{p} for a compact connected Lie group G and so we can assume BX ≃ BG_{p̂}. Hence j : π_{1}(D) ⟶^{≅} π_{1}(X) by Lie theory (cf., [9, §4, no. 6, Prop. 11]), so the same holds for X′.

7. Proof of the corollaries of Theorem 1.2

In this section we prove the Theorems 1.3–1.6 from the introduction. The first theorem is the maximal torus conjecture:

Proof of Theorem 1.3. The proof in the connected case is an extension of [8, Pf. of Thm. 1.10], where a partial result excluding the case p = 2 was given: Assume that (X, BX, e) is a connected finite loop space with a maximal torus i : BT → BX. Let W denote the set of conjugacy classes of self-equivalences φ of BT such that iφ is conjugate to i. It is straightforward to see (consult e.g., [8, Pf. of Thm. 1.10]) that BT_{ p̂} → BX_{p̂} is a maximal torus for the p-compact group X_{p̂} and that 𝔽_{p}-completion allows us to identify (W_{X_{p̂}}, L_{X_{p̂}}) with (W, L ⊗ ℤ_{p}), where L = π_{1}(T). In particular (W, L) is a finite ℤ-reflection group and all reflections have order 2. Furthermore, for a fixed ℤ-reflection group (W, L), there is a bijection between ℤ-root data with underlying ℤ-reflection group (W, L) and ℤ_{2}-root data with underlying ℤ_{2}-reflection group (W, L ⊗ ℤ_{2}) given by the assignments D ↦ D ⊗_{ℤ} ℤ_{2} and (W, L ⊗ ℤ_{2}, {ℤ_{2}b_{σ}}) ↦ (W, L, {L ∩ ℤ_{2}b_{σ}}), as is seen by examining the definitions. Let D be the ℤ-root datum with underlying ℤ-reflection group (W, L) corresponding to D_{X_{ˆ2}}.

By the classification of compact connected Lie groups, cf. [9, §4, no. 9, Prop. 16], there is a (unique) compact connected Lie group G with maximal torus i′ : T → G inducing an isomorphism of ℤ-root data D_{G} ≅ D. The ℤ_{p}-root data of X_{p̂} and G_{p̂} are isomorphic at all primes, since the root data at odd primes are determined by (W, L ⊗ ℤ_{p}). Theorem 1.2 hence implies that for each p, we have a homotopy equivalence φ_{p} : BX_{p̂} → BG_{p̂} such that

\xymatrix{ & BT_{ p̂} \ar[dl]_{i_{p̂}} \ar[dr]^{i′_{p̂}} \\ BX_{p̂} \ar[rr]^{φ_{p}} &  & BG_{p̂}}

commutes. As in [8, Pf. of Thm. 1.10] we see that H^{∗}(BX; ℚ) → H^{∗}(BT; ℚ)^{W} is an isomorphism and since the same is true for BG we also have a map BX_{ℚ} → BG_{ℚ} under BT. We have the following diagram

(7.1)  [figure: ∏pBX_{p̂}; ≃; (∏_{p} BX_{p̂})_{ℚ}; ≃; BX_{ℚ}; ≃; _{∏p}BG_{p̂}; (∏_{p} BG_{p̂})_{ℚ}; BG_{ℚ}.]

The left-hand square in this diagram is homotopy commutative by construction. For the right-hand side note that, since all maps in (7.1) are under BT, the following diagram commutes

\xymatrix{H_{∗}(∏_{p} BX_{p̂}; ℚ) \ar[dd]^{≅} &  & H_{∗}(BX; ℚ) \ar[ll] \ar[dd]^{≅} \\  & H_{∗}(BT; ℚ)_{W} \ar[ul] \ar@{-->}[ur]_{≅} \ar[dr]^{≅} \ar[dl] \\ H_{∗}(∏_{p} BG_{p̂}; ℚ) &  & H_{∗}(BG; ℚ). \ar[ll]}

This implies commutativity on the level of homotopy groups, and since the involved spaces are all products of rational Eilenberg-Mac Lane spaces (since they have homotopy groups only in even degrees) this implies that the diagram (7.1) homotopy commutes. By changing the maps up to homotopy we can hence arrange that the diagram strictly commutes, and taking homotopy pull-backs produces a homotopy equivalence BX → BG, by the Sullivan arithmetic square[10, VI.8.1], as wanted. This proves that every connected finite loop space BX with a maximal torus is homotopy equivalent to BG for some compact connected Lie group G, and in the course of the analysis, we furthermore saw that G is unique (since we can read off the ℤ-root datum of G from BX).

We now give the description of B Aut(BG), also providing a quick description of how the p-completed results are used. By [26, Thm. 1.4] B^{2}Z(G) ≃ B Aut_{1}(BG). (This is a consequence of that B^{2}Z(G_{p̂}) ≃ B Aut_{1}(BG_{p̂}).) By [33, Cor. 3.7] Out(G) ≅ Out(BG). (Since (G/T)^{hT} is homotopically discrete with components the Weyl group, as in [8, Lem. 4.1] and [49, Thm. 2.1], any map f : BG → BG gives rise to a map φ : BT → BT over f, unique up to Weyl group conjugation and since φ_{p̂} ∈ Aut(D_{G_{p̂}}) for all p one sees that φ ∈ Aut(D_{G}); now φ determines the collection {φ_{p̂}} which by p-complete results determines {f_{ p̂}}, which again determines f by the arithmetic square.) Finally by a theorem of de Siebenthal [15, Ch. I, §2, no. 2] (see also [9, §4, no. 10]), the short exact sequence 1 → G/Z(G) → Aut(G) → Out(G) → 1 is split, so the fibration B Aut(BG) → B Out(BG) has a section. (See also [4, §6].) Taken together these facts establish that B Aut(BG) ≃ (B^{2}Z(G))_{h Out(G)} as claimed.

The non-connected case follows easily from the connected, using our knowledge of selfmaps of classifying spaces of compact connected Lie groups: Suppose that X is potentially non-connected, let X_{1} be the identity component, and π the component group. Note that homotopy classes of spaces Y such that P_{1}(Y) is homotopy equivalent to Bπ and the fiber Y ⟨1⟩ is homotopy equivalent to BX_{1} can alternatively be described as homotopy classes of fibrations Z → Y → K, with Z is homotopic to BX_{1} and K is homotopic to Bπ. (A homotopy equivalence of fibrations means a compatible triple of homotopy equivalences between fibers, total spaces, and base spaces.)

By the first part of the proof we know that BX_{1} ≃ BG_{1} for a unique compact connected Lie group G_{1}. Homotopy classes of fibrations with base homotopy equivalent to Bπ and fiber homotopy equivalent to BX_{1} are classified by Out(π)-orbits on the set of free homotopy classes [Bπ, B Aut(BG_{1})]. By the results in the connected case the space B Aut(BG_{1}) sits in a split fibration

B^{2}Z(G_{1}) → B Aut(BG_{1}) → B Out(G_{1}).

Hence Out(π)-orbits on the set [Bπ, B Aut(BG_{1})] correspond to (Out(π) × Out(G_{1}))-orbits on the set

∐_{α∈Hom(π,Out(G_{1}))} H_{α}^{2}(π; Z(G_{1})).

This agrees with the classification of isomorphism classes of group extensions of the form 1 → H → ? → K → 1, where H is isomorphic to G_{1} and K is isomorphic to π. (An isomorphism of group extensions is here a compatible triple of isomorphisms.) Since the identity component H is necessarily a characteristic subgroup, isomorphism classes of group extensions as above are in one-to-one correspondence with isomorphism classes of compact Lie groups with identity component isomorphic to G_{1} and component group isomorphic to π. These equivalences put together give that there is a one-to-one correspondence between homotopy classes of spaces Y such that P_{1}(Y) ≃ Bπ and Y ⟨1⟩ ≃ BG_{1} and isomorphism classes of compact Lie groups with identity component isomorphic to G_{1} and component group isomorphic to π. Hence our BX is homotopy equivalent to BG for a unique compact Lie group G, completing the proof of the theorem. □

Proof of Theorem 1.4. Let Y be a space such that H^{∗}(Y; 𝔽_{2}) is a graded polynomial algebra of finite type. Let V = H_{1}(Y; 𝔽_{2}) (dual to H^{1}(Y; 𝔽_{2})) and let Y′ denote the fiber of the classifying map Y → BV. Clearly Y′ is connected and since π_{1}(BV) is a finite 2group it follows from [17] that the Eilenberg-Moore spectral sequence for the fibration Y′ → Y → BV converges strongly to H^{∗}(Y′; 𝔽_{2}). The map H^{∗}(BV; 𝔽_{2}) → H^{∗}(Y; 𝔽_{2}) is an isomorphism in degree 1 and hence injective since H^{∗}(Y; 𝔽_{2}) is a polynomial algebra, so H^{∗}(Y; 𝔽_{2}) is free over H^{∗}(BV; 𝔽_{2}). Hence the spectral sequence collapses and we get an isomorphism of rings (but not necessarily of algebras over the Steenrod algebra) H^{∗}(Y; 𝔽_{2}) ≅ H^{∗}(Y′; 𝔽_{2}) ⊗ H^{∗}(BV; 𝔽_{2}). In particular H_{1}(Y′; 𝔽_{2}) = 0 so by [10, Prop. VII.3.2] Y′ is 𝔽_{2}- good and π_{1}(Y′_{ˆ2}) = 0. So to prove the theorem, we can without restriction assume that Y′ is 𝔽_{2}-complete and simply connected.

Write π_{2}(Y′) ≅ F ⊕ T, where F is a finitely generated free ℤ_{2}-module and T is a finite abelian 2-group, and let Y′′ be the fiber of the map Y′ → B^{2}F. The induced homomorphism H_{2}(Y′; 𝔽_{2}) → H_{2}(B^{2}F; 𝔽_{2}) is an epimorphism so H^{∗}(B^{2}F; 𝔽_{2}) → H^{∗}(Y′; 𝔽_{2}) is injective. As above we obtain an isomorphism H^{∗}(Y′; 𝔽_{2}) ≅ H^{∗}(Y′′; 𝔽_{2}) ⊗ H^{∗}(B^{2}F; 𝔽_{2}) as rings.

By construction Y′′ is simply connected, π_{2}(Y′′) is finite, and by the fibre lemma [10, Lem. II.5.1] Y′′ is 𝔽_{2}-complete. Since H^{∗}(Y′′; 𝔽_{2}) is polynomial, the Eilenberg-Moore spectral sequence shows that H^{∗}(ΩY′′; 𝔽_{2}) is 𝔽_{2}-finite, so Y′′ is the classifying space of a connected 2-compact group. The first part of Theorem 1.4 now follows from the classification Theorem 1.1. The second part follows from this using the calculation of the mod 2 cohomology of the simple simply connected Lie groups, cf. [36, Thm. 5.2]. □

Remark 7.1. In addition to the list in Theorem 1.4, the only polynomial rings arising as H^{∗}(BG; 𝔽_{2}) for a simple compact connected Lie group G are 𝔽_{2}[x_{2}, x_{3}, …, x_{n}] for G = SO(n), n ≥ 5, and 𝔽_{2}[x_{2}, x_{3}, x_{8}, x_{12}, …, x_{8n+4}] for G = PSp(2n + 1), n ≥ 0, cf. [36, Thm. 5.2]. It is conceivable that any graded polynomial algebra of finite type which is the mod 2 cohomology ring of a space, is a tensor product of these factors and the ones listed in Theorem 1.4.

Proof of Theorem 1.5. The first statement claims that a polynomial 𝔽_{2}-algebra A^{∗} with given action of the Steenrod algebra A_{2}, can be realized by at most one space Y, up to 𝔽_{2}-equivalence, if A^{∗} has all generators in degree ≥ 3. For this notice that, as in the proof of Theorem 1.4, the assumptions assure that Y _{ ˆ2} ≃ BX for a simply connected 2-compact group X. Using the classification Theorem 1.1, the statement can now easily be checked as done in Proposition 7.2 below.

We now prove the second statement, that for any polynomial 𝔽_{2}-algebra P^{∗}, there are only finitely many spaces Y, up to 𝔽_{2}-equivalence, with H^{∗}(Y; 𝔽_{2}) ≅ P^{∗} as rings. By the proof of Theorem 1.4, we can assume that Y is 2-complete, and any such Y fits in a fibration sequence Y′ → Y → BV where V = H_{1}(Y; 𝔽_{2}). It also follows that H^{∗}(Y′; 𝔽_{2}) is a polynomial ring, uniquely determined as an algebra over the Steenrod algebra from H^{∗}(Y; 𝔽_{2}). In particular Y′ ≃ BX for a connected 2-compact group X. By Proposition 8.18, Out(D_{X}) only contains finitely many 2-subgroups up to conjugation. Hence the description of B Aut(BX) in Theorem 1.2 implies that [BV, B Aut(BX)] is finite, so it is enough to see that there are only a finite number of possibilities for Y′ given H^{∗}(Y′; 𝔽_{2}) as an algebra over the Steenrod algebra. This again follows easily from the classification of 2-compact groups: The rank of X is bounded above by the Krull dimension of the cohomology ring, so by the classification of p-compact groups, Theorem 1.2, it is hence enough to see that there are only a finite number of ℤ_{2}-root data with rank less than a fixed rank. This is the result of Proposition 8.17. □

We now give a proof of the auxiliary uniqueness result referred to in the proof of Theorem 1.5.

Proposition 7.2. Suppose X is a 2-compact group of the form BX ≃ BG_{ˆ2}× BDI(4)^{s}, for a simply connected compact Lie group G and s ≥ 0, such that H^{∗}(BX; 𝔽_{2}) is a polynomial algebra. Then X has a unique maximal elementary abelian 2-subgroup ν : BE → BX, and the Weyl group (W (ν), E) together with the homomorphism H^{6}(BX; 𝔽_{2}) → H^{6}(BE; 𝔽_{2}) is an invariant of H^{∗}(BX; 𝔽_{2}) as an algebra over the Steenrod algebra A_{2}, which uniquely determines BX up to homotopy equivalence. In particular BX is uniquely determined up to homotopy equivalence by H^{∗}(BX; 𝔽_{2}) as an A_{2}-algebra.

Proof. By Lannes theory [37, Thm. 0.4] homotopy classes of maps from classifying spaces of elementary abelian 2-groups to BX are determined by H^{∗}(BX; 𝔽_{2}) as an algebra over the Steenrod algebra. Furthermore, the fact that H^{∗}(BX; 𝔽_{2}) is assumed to be a polynomial algebra, guarantees that there is only one maximal elementary abelian 2-subgroup ν : BE → BX, up to conjugation, by [54, Cor. 10.7] together with the fact that this is true for B DI(4). (This can also be deduced using the unstable algebra techniques of [2] and [31], or simply by inspecting the calculations of Griess [30] below.) Let (W (ν), E) denote its Weyl group, which also only depends on H^{∗}(BX; 𝔽_{2}), and we view this as a pair with W (ν) a subgroup of GL(E). By [36, Thm. 5.2], G is a direct product of the groups SU(n), Sp(n), Spin(7), Spin(8), Spin(9), G_{2} and F_{4}. In these cases, if E is the maximal elementary abelian 2subgroup of G, W_{G_{ˆ2}}(ν) = N_{G}(E)/C_{G}(E) whose structure is well-known in these cases, e.g., by computations of Griess [30, §5, Thm. 6.1 and Thm. 7.3]. (See also [63, Prop. 3.2] for details on the cases Spin(8) ⊆ Spin(9) ⊆ F_{4}.) Since W_{DI(4)}(ν) = GL_{4}(𝔽_{2}) by construction [24], it follows that the group (W (ν), E) is a direct product of the following (with matrix groups acting on columns):

W_{SU(n)}(ν) = (Σ_{n}, V′_{n−1}), W_{Sp(n)}(ν) = (Σ_{n}, V_{n}),

W_{G_{2}}(ν) = GL_{3}(𝔽_{2}), W_{DI(4)}(ν) = GL_{4}(𝔽_{2}),

W_{Spin(7)}(ν) = [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & GL_{3}(𝔽_{2}) \end{matrix} ] , W_{Spin(8)}(ν) = [ \begin{matrix}1 0 0 1 & \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \end{matrix} \\ 0 0 0 0 0 0 & GL_{3}(𝔽_{2}) \end{matrix} ] ,

W_{Spin(9)}(ν) = [ \begin{matrix}\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} & \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \end{matrix} \\ 0 0 0 0 0 0 & GL_{3}(𝔽_{2}) \end{matrix} ] , W_{F_{4}}(ν) = [ \begin{matrix}GL_{2}(𝔽_{2}) & \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \end{matrix} \\ 0 0 0 0 0 0 & GL_{3}(𝔽_{2}) \end{matrix} ] .

Here V_{n} is the n-dimensional permutation module for 𝔽_{2}[Σ_{n}] and V′_{n−1} is the (n − 1)- dimensional submodule consisting of elements with coordinate sum 0. The Weyl group W_{Sp(n)}(ν) = (Σ_{n}, V_{n}) decomposes as (Σ_{n}, V′_{n−1}) × (1, L), L = V_{n}^{Σ_{n}} ≅ 𝔽_{2}, when n is odd, n ≥ 3. However, after this decomposition, all the listed pairs satisfy that V is indecomposable as an 𝔽_{2}[W]-module. (Note that this is a priori stronger than just saying that (W, V) does not split as a product; however, it follows from unstable algebra techniques [48, Secs. 5 and 7] that (W, V) is an 𝔽_{2}-reflection group, and hence the two notions are actually equivalent.) By the Krull-Schmidt theorem [14, 6.12(ii)] this decomposition V = V_{1} ⊕ … ⊕ V_{m} as 𝔽_{2}[W]-modules is unique up to permutation, and since W_{i} is characterized as the pointwise stabilizer of V_{1} ⊕ … ⊕ V̂_{i} ⊕ … ⊕ V_{m}, we get that the decomposition of (W, V) as a product is unique as well, up to permutation. Thus the structure of (W (ν), E) as a product of finite indecomposable groups, is an invariant which almost characterizes BX up to homotopy equivalence, except that for n odd, n ≥ 3, the group (Σ_{n}, V′_{n−1}) arises from both SU(n)_{ˆ2} and Sp(n)_{ˆ2}. However since H^{6}(B SU(n); 𝔽_{2}) → H^{6}(BV′_{n−1}; 𝔽_{2}) is injective and H^{6}(B Sp(n); 𝔽_{2}) → H^{6}(BV′_{n−1}; 𝔽_{2}) is trivial, we conclude that (W (ν), E) together with the homomorphism H^{6}(BX; 𝔽_{2}) → H^{6}(BE; 𝔽_{2}) characterizes BX up to homotopy equivalence. This proves the proposition since both data are determined by the A_{2}-action on H^{∗}(BX; 𝔽_{2}). □

Proof of Theorem 1.6. It is obvious that there is a one-to-one correspondence between isomorphism classes of p-compact groups with identity component isomorphic to X_{1} and component group isomorphic to π, and equivalence classes of fibration sequences F → E ⟶^{p} B with F homotopy equivalent to BX_{1} and B homotopy equivalent to Bπ. It is likewise obvious that in this case B Aut(E) is homotopy equivalent to B Aut(p), where Aut(p) is the space of self-homotopy equivalences of the fibration p.

By the classification of fibrations (see [20]), equivalence classes of such fibrations are in one-to-one correspondence with Out(B)-orbits on [B, B Aut(F)], and the space B Aut(p) equals (map(B, B Aut(F))_{C(p)})_{h Aut(B)}, where C(p) denotes the Out(B)-orbit on [B, B Aut(F)] of the element classifying p : E → B.

The above considerations completely reduces the proof of the theorem, to our classification theorem for connected p-compact groups, Theorem 1.2, except for the finiteness statement. For this note that Proposition 8.18 implies that [Bπ, B Out(D)] is finite so the finiteness of [Bπ, B aut(D)] follows. □

Remark 7.3. As stated in the introduction, our classification also shows that Bott’s theorem on the cohomology of X/T, the Peter-Weyl theorem, as well as Borel’s characterization of when centralizer of elements of order p are connected, stated as Theorems 1.5, 1.6 and 1.9 in [8], hold verbatim for 2-compact groups. To prove these results it suffices by Theorem 1.1 to check them for DI(4), since they are well known for compact Lie groups. For DI(4) one argues as follows: Bott’s theorem [8, Thm. 1.5] follow from [24, Thm. 1.8(2)], the Peter-Weyl theorem [8, Thm. 1.6] is a result of Ziemiański [67], and it is trivial to check that [8, Thm. 1.9] hold.

8. Appendix: Properties of ℤ_{p}-root data

The purpose of this section is to establish some general results about ℤ_{p}-root data of p-compact groups, needed in the proof of the main theorem. The analogous results for ℤ-root data and compact Lie groups are often well known; see [9, 16]. We build on the paper [28] by Dwyer-Wilkerson and our earlier paper [4].

We briefly recall the definition of root data from the introduction: For an integral domain R, an R-reflection group is a pair (W, L) where L is a finitely generated free R-module and W is a subgroup of Aut_{R}(L) generated by reflections (i.e. elements σ ∈ Aut_{R}(L) such that 1 − σ ∈ End_{R}(L) has rank one). If R is a principal ideal domain, we define an R-root datum to be a triple D = (W, L, {Rb_{σ}}) where (W, L) is a finite R-reflection group and for each reflection σ ∈ W, Rb_{σ} is a rank one submodule of L with im(1 − σ) ⊆ Rb_{σ} and w(Rb_{σ}) = Rb_{wσw^{−1}} for all w ∈ W. If R → R′ is a monomorphism of integral domains, and D an R-root datum, we can define an R′-root datum by D⊗_{R} R′ = (W, L⊗_{R} R′, {Rb_{σ} ⊗_{R} R′}).

The element b_{σ}, defined up to a unit in R, is called the coroot associated to σ. By definition b_{σ} determines a unique linear map β_{σ} : L → R called the associated root such that σ(x) = x + β_{σ}(x)b_{σ} for x ∈ L. Define the coroot lattice L_{0} ⊆ L as the sublattice spanned by the coroots b_{σ} and the fundamental group of D by π_{1}(D) = L/L_{0}. In general Rb_{σ} ⊆ ker(N), where N = 1+σ+…+σ^{|σ|−1} is the norm element (cf. the proof of Lemma 8.8 below), so giving an R-root datum with underlying reflection group (W, L) corresponds to choosing a cyclic R-submodule of H^{1}(⟨σ⟩; L) for each conjugacy class of reflections σ. In particular for R = ℤ_{p}, p odd, the notions of a ℤ_{p}-reflection group and a ℤ_{p}-root datum agree. If R has characteristic zero, an R-root datum, or an R-reflection group, is called irreducible if the representation W → GL(L ⊗_{R} K) is irreducible, where K denotes the quotient field of R, and it is said to be exotic if furthermore the values of the character of this representation are not all contained in ℚ.

We are now ready to state the classification of ℤ_{p}-root data, which follows easily from the classification of finite ℤ_{p}-reflection groups [8, Thm. 11.1]. This classification is again based on the classification of finite ℚ_{p}-reflection groups [13] [22] which states that for a fixed prime p, isomorphism classes of finite irreducible ℚ_{p}-reflection groups are in natural one-to-one correspondence with isomorphism classes of finite irreducible C-reflection groups (W, V) [55] for which the values of the character of W → GL(V) are embeddable in ℚ_{p}; see e.g., [3, Table 1] for an explicit list of groups and primes.

Theorem 8.1 (The classification of ℤ_{p}-root data; splitting version). (1) Any ℤ_{p}-root datum is isomorphic to a ℤ_{p}-root datum of the form (D_{1} ⊗_{ℤ} ℤ_{p}) × D_{2}, where D_{1} is a ℤ-root datum and D_{2} is a direct product of exotic ℤ_{p}-root data.

(2) There is a one-to-one correspondence between isomorphism classes of exotic ℤ_{p}-root data and isomorphism classes of exotic ℚ_{p}-reflection groups given by the assignment D = (W, L, {ℤ_{p}b_{σ}}) ⇝ (W, L ⊗_{ℤ_{p}} ℚ_{p}). Moreover π_{1}(D) = 0 for any exotic ℤ_{p}-root datum D.

Proof. For any ℤ_{p}-root datum D = (W, L, {ℤ_{p}b_{σ}}), [8, Thm. 11.1] gives a splitting (W, L) ≅ (W_{1}, L_{1} ⊗_{ℤ} ℤ_{p}) × (W_{2}, L_{2}) of (W, L), where (W_{1}, L_{1}) is a finite ℤ-reflection group and (W_{2}, L_{2}) is a direct product of exotic ℤ_{p}-reflection groups. It follows by definition that there are unique ℤ_{p}-root data D′ and D_{2} with underlying reflection groups (W_{1}, L_{1} ⊗_{ℤ} ℤ_{p}) and (W_{2}, L_{2}) such that D ≅ D′ × D_{2}, and by the same argument D_{2} splits as a direct product of exotic ℤ_{p}-root data. Furthermore, writing D′ = (W_{1}, L_{1} ⊗_{ℤ} ℤ_{p}, {ℤ_{p}b_{σ}}) it is clear that D′ ≅ D_{1} ⊗_{ℤ} ℤ_{p} where D_{1} = (W_{1}, L_{1}, {L_{1} ∩ ℤ_{p}b_{σ}}). This proves (1).

By [8, Thm. 11.1], the assignment (W, L) ⇝ (W, L⊗_{ℤ_{p}}ℚ_{p}) establishes a one-to-one correspondence between exotic ℤ_{p}-reflection groups up to isomorphism and exotic ℚ_{p}-reflection groups up to isomorphism. To prove the first part of (2) it thus suffices to show that any exotic ℤ_{p}-reflection group (W, L) can be given a unique ℤ_{p}-root datum structure. For p > 2 this holds since H^{1}(⟨σ⟩; L) = 0, cf. the discussion in the beginning of this section. For p = 2, (W, L) ≅ (W_{DI(4)}, L_{DI(4)}) where the claim follows by direct inspection (cf. [28, Rem. 7.2]).

For any ℤ_{p}-root datum D = (W, L, {ℤ_{p}b_{σ}}), the formula σ(x) = x + β_{σ}(x)b_{σ} shows that the coroot lattice L_{0} contains the lattice spanned by the elements (1−w)(x), w ∈ W, x ∈ L. Hence the final claim follows from the fact that H_{0}(W; L) = 0 for any exotic ℤ_{p}-reflection group (W, L) [8, Thm. 11.1]. □

8.1. Root datum, normalizer extension, and root subgroups of a p-compact group. For any connected p-compact group X with maximal torus T, the Weyl group W_{X} acts naturally on L_{X} = π_{1}(T) as a finite ℤ_{p}-reflection group [25, Thm. 9.7(ii)]. For p odd, H^{1}(⟨σ⟩; L) = 0 for any reflection σ, so the finite ℤ_{p}-reflection group (W_{X}, L_{X}) gives rise to a unique ℤ_{p}-root datum D_{X}. The construction of root data for connected 2-compact groups, in the present form, is due to Dwyer-Wilkerson [28, §9]: Let T̆ be the discrete approximation to T, N̆_{X} the discrete approximation to the maximal torus normalizer N_{X} and σ ∈ W_{X} a reflection. Define T̆^{+}(σ) = T̆^{⟨σ⟩} and let T̆_{0}^{+}(σ) denote its maximal divisible subgroup. Then X(σ) = C_{X}(T̆_{0}^{+}(σ)) is a connected 2-compact group with Weyl group ⟨σ⟩ and N̆ (σ) = CN̆_{X} (T̆_{0}^{+}(σ)) is a discrete approximation to its maximal torus normalizer.

Furthermore, let T̆_{0}^{−}(σ) denote the maximal divisible subgroup of T̆^{−}(σ) = ker(T̆ ⟶^{1+σ} T̆) and define the root subgroup N̆_{X,σ} by

N̆_{X,σ} = {x ∈ N̆ (σ) | ∃ y ∈ T̆_{0}^{−}(σ) : x is conjugate to y in X(σ)}.

Then there is a short exact sequence

(8.1)  1 → T̆_{0}^{−}(σ) → N̆_{X,σ} → ⟨σ⟩ → 1,

and we define

ℤ_{2}b_{σ} = \begin{cases}im(L_{X} ⟶^{1−σ} L_{X}) & if (8.1) splits, \\ ker(L_{X} ⟶^{1+σ} L_{X}) & otherwise. \end{cases}

The root datum of X is then the ℤ_{2}-root datum D_{X} = (W_{X}, L_{X}, {ℤ_{2}b_{σ}}); see [28, §6 and 9] and [4] for a further discussion.

Conversely, the maximal torus normalizer and the root subgroups of a connected pcompact group can be reconstructed from its root datum: For a ℤ_{p}-root datum D = (W, L, {ℤ_{p}b_{σ}}) there is [28, Def. 6.15] [4, §3] an algebraically defined extension

1 → T̆ → N̆_{D} → W → 1

called the normalizer extension with a subextension 1 → T̆_{0}^{−}(σ) → N̆_{D,σ} → ⟨σ⟩ → 1 for each reflection σ ∈ W. For a connected p-compact group X with ℤ_{p}-root datum D_{X} there is an isomorphism of extensions [28, Prop. 1.10] [4, Thm. 3.1(2)]

[figure: 1; T̆; N̆_{D}; ≅; W; 1; 1; T̆; N̆_{X}; W; 1]

sending the root subgroups N̆_{D,σ} to N̆_{X,σ} for all reflections σ, and any such isomorphism is unique up to conjugation by an element in T̆.

We define BN_{D} as the fiber-wise 𝔽_{p}-completion [10, Ch. I, §8] of BN̆_{D} and likewise introduce the (non-discrete) root subgroups BN_{X,σ} and BN_{D,σ} by fiber-wise 𝔽_{p}-completion of the corresponding discrete versions BN̆_{X,σ} and BN̆_{D,σ}.

Recollection 8.2 (The Adams-Mahmud map). By [8, Lem. 4.1] we have an “Adams-Mahmud” homomorphism Φ : Out(BX) → Out(N̆), given by associating to f : BX → BX the homomorphism Φ(f) : N̆ → N̆, unique up to conjugation, such that the diagram

\xymatrix{BN̆ \ar[r]^{BΦ(f)} \ar[d] & BN̆ \ar[d] \\ BX \ar[r]^{f} & BX}

commutes up to homotopy.

By [4, Thm. B], Φ factors through Out(N̆ , {N̆_{σ}}) = {[φ] ∈ Out(N̆) | φ(N̆_{σ}) = N̆_{φ(σ)}}, which is isomorphic to Out(D_{X}) via restriction to T̆. We again denote this map by Φ. Likewise, as e.g., explained in [8, Prop. 5.1], fiber-wise 𝔽_{p}-completion [10, Ch. I, §8] induces a natural isomorphism Out(N̆) ⟶^{≅} Out(BN), and we can hence equivalently view Φ(f) as an element in Out(BN), and we will not notationally distinguish between the two cases. We denote the subgroup of Out(BN) corresponding to Out(N̆ , {N̆_{σ}}) by Out(BN , {BN_{σ}}).

8.2. Centers and fundamental groups. If D = (W, L, {ℤ_{p}b_{σ}}) is a ℤ_{p}-root datum, we define a subdatum of D to be a ℤ_{p}-root datum of the form (W′, L, {ℤ_{p}b_{σ}}_{σ∈Σ′}) where (W′, L) is a reflection subgroup of (W, L) and Σ′ is the set of reflections in W′. For the next result, recall [26, Def. 4.1] that a homomorphism f : BX → BY is called a monomorphism of maximal rank if the homotopy fiber Y/X is 𝔽_{p}-finite and X and Y have the same rank.

Proposition 8.3. Let X and Y be connected p-compact groups. If f : BX → BY is a monomorphism of maximal rank then D_{X} naturally identifies with a subdatum of D_{Y}.

Proof. By definition there is a maximal torus i : BT → BX such that f ∘ i : BT → BY is a maximal torus for Y. Thus we can identify L_{X} = L_{Y} = π_{1}(T) and by [26, Lem. 4.4] we have an induced monomorphism W_{X} → W_{Y}. This proves the result for p odd, since in that case the ℤ_{p}-root data D_{X} and D_{Y} are uniquely determined by their underlying reflection groups (W_{X}, L_{X}) and (W_{Y} , L_{Y}). In the case p = 2 the result follows from the construction of the ℤ_{2}-root data of X and Y, cf. [28, Pf. of Lem. 9.16]. □

For a ℤ_{p}-root datum D = (W, L, {ℤ_{p}b_{σ}}), we let T̆ = L ⊗_{ℤ_{p}} ℤ/p^{∞} be the associated discrete torus and for a reflection σ ∈ W, we define h_{σ} = b_{σ} ⊗ \frac{1}{2} ∈ T̆. Clearly h_{σ} is independent of the choice of b_{σ} and conversely h_{σ} determines ℤ_{p}b_{σ}, cf. [28, §2 and §6]. So instead of (W, L, {ℤ_{p}b_{σ}}) we might as well work with (W, T̆, {h_{σ}}); we will use these two viewpoints interchangeably without further mention. Also note that h_{σ} = 1 for p odd. When σ ∈ W is a reflection, we define the singular set S(σ) by

S(σ) = ⟨T̆_{0}^{+}(σ), h_{σ}⟩ = ker(β_{σ} ⊗_{ℤ_{p}} ℤ/p^{∞} : T̆ → ℤ/p^{∞}),

cf. [26, Def. 7.3] and [4, Pf. of Prop. 4.2]. Define the discrete center Z̆(D) of D as ⋂_{σ} S(σ), where the intersection is taken over all reflections σ ∈ W. In other words, letting M_{0} denote the root lattice, i.e., the ℤ_{p}-sublattice of L^{∗} spanned by the roots β_{σ}, we have the identification

(8.2)  Z̆(D) = ker (T̆ = Hom_{ℤ_{p}}(L^{∗}, ℤ/p^{∞}) ↠ Hom_{ℤ_{p}}(M_{0}, ℤ/p^{∞})) .

The following proposition translates into the present language the results of Dwyer-Wilkerson [26, §7] on how to compute the center and centralizers of toral subgroups of X.

Proposition 8.4. Let X be a connected p-compact group with ℤ_{p}-root datum D_{X} = (W, L, {ℤ_{p}b_{σ}}).

(1) The center BZ(X) of X is canonically homotopy equivalent to the center BZ(D) =

(BZ̆(D))_{p̂} of D.

(2) The identity component Z(X)_{1} of the center has ℤ_{p}-root datum (1, L^{W} , ∅).

(3) For A ⊆ T̆, let W (A) be the pointwise stabilizer of A in W, Σ_{A} the set of reflections σ with A ⊆ S(σ), and W (A)_{1} the subgroup of W generated by Σ_{A}. Then the centralizer C_{X}(A) has Weyl group W (A) and its identity component C_{X}(A)_{1} has ℤ_{p}-root datum equal to the subdatum D_{A} = (W (A)_{1}, L, {ℤ_{p}b_{σ}}_{σ∈Σ_{A}}) of D.

Proof. The first part is [26, Thm. 7.5]. The second part follows easily from this since the maximal divisible subgroup of Z̆(D) equals

⋂_{σ} T̆_{0}^{+}(σ) = ⋂_{σ} (L^{+}(σ) ⊗_{ℤ_{p}} ℤ/p^{∞}) = L^{W} ⊗_{ℤ_{p}} ℤ/p^{∞}.

Part (3) follows by combining [26, Thm. 7.6] and Proposition 8.3 since BC_{X}(A)_{1} → BX is a monomorphism of maximal rank [26, Prop. 4.3]. □

Let D = (W, L, {ℤ_{p}b_{σ}}) be a ℤ_{p}-root datum. Recall that the coroot lattice L_{0} ⊆ L is the ℤ_{p}-lattice spanned by the coroots b_{σ} and that the fundamental group π_{1}(D) is the quotient L/L_{0}. Define H_{n}^{ℤ_{p}}(X) = lim_{k} H_{n}(X; ℤ/p^{k}). The following proposition, which refines [8, Prop. 10.2], constructs a canonical epimorphism π_{1}(D_{X}) → π_{1}(X) which will be shown to be an isomorphism in Theorem 8.6 below.

Proposition 8.5. Let X be a connected p-compact group with maximal torus T and ℤ_{p}- root datum D_{X} = (W, L, {ℤ_{p}b_{σ}}). Then the homomorphism L = H_{2}^{ℤ_{p}}(BT) → H_{2}^{ℤ_{p}}(BX) ≅ π_{1}(X) factors through π_{1}(D_{X}) and the induced homomorphism π_{1}(D_{X}) → π_{1}(X) is surjective with finite kernel.

Proof. For p odd we have im(1 − σ) = ℤ_{p}b_{σ} for all σ, so L/L_{0} ≅ H_{0}(W; L) and the result follows from [8, Prop. 10.2].

Assume now that p = 2 and let σ ∈ W be a reflection. To see the first part we have to show that the homomorphism L = H_{2}^{ℤ_{2}}(BT) → H_{2}^{ℤ_{2}}(BX) ≅ π_{1}(X) vanishes on the coroots b_{σ}. This follows from the construction of the root datum of X: X(σ) = C_{X}(T̆_{0}^{+}(σ)) is a connected 2-compact group and Proposition 8.4(3) shows that D_{X(σ)} equals the subdatum (⟨σ⟩ , L, {ℤ_{2}b_{σ}}) of D_{X}. The commutative diagram

[figure: L = H_{2}^{ℤ_{2}}(BT); H_{2}^{ℤ_{2}}(BX(σ)) ≅ π_{1}(X(σ)); L = H_{2}^{ℤ_{2}}(BT); H_{2}^{ℤ_{2}}(BX) ≅ π_{1}(X)]

shows that it suffices to prove the claim for X(σ). However since X(σ) is a connected 2-compact group of rank 1 it follows (cf. [28, p. 1369–1370]) that X(σ) ≅ G_{ˆ2} for G = SU(2) × (S^{1})^{r−1}, SO(3) × (S^{1})^{r−1} or U(2) × (S^{1})^{r−2} where r is the rank of X. The wellknown formula for the fundamental group of a compact connected Lie group in terms of its root datum, cf. [9, §4, no. 6, Prop. 11] or [1, Thm. 5.47], now established the first part of the proposition.

Since im(1 − σ) ⊆ ℤ_{2}b_{σ} by definition, L → L/L_{0} = π_{1}(D_{X}) factors through H_{0}(W; L), so the final claim now follows from [8, Prop. 10.2]. □

We will also be using the following formula for the fundamental group of a p-compact group, proved by Dwyer-Wilkerson [29] by a transfer argument as this paper was being written (see also [8, Rem. 10.3]). The formula was previously known for p odd, by our classification [8], and we sketch in Remark 6.2 how one can bypass the use of this formula also in the classification for p = 2 by a more cumbersome argument which we had originally envisioned using in this paper; in particular providing an independent proof.

Theorem 8.6 (Dwyer-Wilkerson [29] and Remark 6.2). Let X be a connected p-compact group. Then π_{1}(D_{X}) ⟶^{≅} π_{1}(X) induced by the maximal torus T → X.

Proof. By Theorem 8.1(1) we may write D_{X} = D_{1} × D_{2}, where D_{1} is of the form D′ ⊗_{ℤ} ℤ_{p} for a ℤ-root datum D′, and D_{2} is a direct product of exotic ℤ_{p}-root data. By [27, Thm. 1.4] this induces a splitting BX ≃ BX_{1} × BX_{2} with D_{X_{i}} ≅ D_{i}. We have to show that the kernel of L = π_{1}(T) → π_{1}(X) equals the coroot lattice L_{0}; by the above it suffices to treat the case where D_{X} is exotic and the case where D_{X} is of the form D′ ⊗_{ℤ} ℤ_{p} for a ℤ-root datum D′.

In the first case, Theorem 8.1(2) shows that π_{1}(D_{X}) = 0 so the result follows from Proposition 8.5.

In the second case we have D_{X} ≅ D_{G_{p̂}} for some compact connected Lie group G. By the result of Dwyer-Wilkerson [29, Thm. 1.1] the kernel of L = π_{1}(T) → π_{1}(X) equals the kernel of H_{2}^{ℤ_{p}}(BT_{X}) → H_{2}^{ℤ_{p}}(BN_{X}). Since the maximal torus normalizer may be reconstructed from the root datum by [28, Prop. 1.10] for p = 2 and [3, Thm. 1.2] for p odd, we may identify the homomorphism H_{2}^{ℤ_{p}}(BT_{X}) → H_{2}^{ℤ_{p}}(BN_{X}) with the homomorphism H_{2}(BT_{G}; ℤ) → H_{2}(BN_{G}; ℤ) tensored by ℤ_{p}. The result now follows from the corresponding result for compact Lie groups, cf. [9, §4, no. 6, Prop. 11] or [1, Thm. 5.47]. □

8.3. Covers and quotients. We now start to address how the root datum behaves upon taking covers and quotients of a p-compact group.

Lemma 8.7. Let f : BX → BY be a monomorphism of maximal rank between connected p-compact groups X and Y. If π_{1}(D_{Y}) = 0 then BZ(X) ≃ BZ(N_{X}).

Proof. For p odd, the conclusion holds for any connected p-compact group X [26, Rem. 7.7], so we may suppose p = 2. Since π_{1}(D_{Y}) = 0, D_{Y} does not have any direct factors isomorphic to D_{SO(2n+1)_{ˆ2}}, so [4, §5] implies that the singular set S_{Y} (σ) with respect to Y equals T̆^{+}(σ) for any reflection σ ∈ W_{Y}. By Proposition 8.3, D_{X} identifies with a subdatum of D_{Y} and hence S_{X}(σ) = T̆^{+}(σ) for all reflections σ ∈ W_{X}. The claim now follows from Proposition 8.4(1). □

Lemma 8.8. Let D = (W, L, {ℤ_{p}b_{σ}}) be a ℤ_{p}-root datum with coroot lattice L_{0}. Then L_{0} ∩ L^{W} = 0 and L_{0} ⊕ L^{W} has finite index in L. In particular W acts faithfully on L_{0}.

Proof. The ℚ_{p}[W]-module V = L ⊗_{ℤ_{p}} ℚ_{p} decomposes as V = V^{W} ⊕ U where U^{W} = 0. Writing b_{σ} = x + y with x ∈ V^{W} and y ∈ U we have σ(b_{σ}) = x + σ(y) and hence σ(b_{σ}) − b_{σ} ∈ U. Also σ(b_{σ}) ≠ b_{σ}: If N = 1 + σ + … + σ^{|σ|−1} is the norm element, then β_{σ}(x)N(b_{σ}) = N(σ − 1)(x) = 0 for all x ∈ L. Since σ ≠ 1 we have β_{σ} ≠ 0 so Nb_{σ} = 0. Thus σ(b_{σ}) ≠ b_{σ} since otherwise Nb_{σ} = |σ|b_{σ} ≠ 0. This proves that (σ − 1)(b_{σ}) = rb_{σ} with r ≠ 0 so b_{σ} ∈ U. Thus L_{0} ⊆ U and L_{0} ∩ L^{W} = 0 as desired.

Since |W |x = (∑_{w∈W} wx) + ∑_{w∈W} (x − wx) ∈ L^{W} + L_{0}, for any x ∈ L we see that L_{0} ⊕ L^{W} has finite index in L, and in particular W acts faithfully on L_{0}. □

Let D = (W, L, {ℤ_{p}b_{σ}}) be a ℤ_{p}-root datum and let L_{0} be the coroot lattice. If L′ is a ℤ_{p}-lattice with L_{0} ⊆ L′ ⊆ L, the formula σ(x) = x + β_{σ}(x)b_{σ} shows that L′ is W -invariant. By Lemma 8.8, W acts faithfully on L_{0} and hence also on L′, so (W, L′, {ℤ_{p}b_{σ}}) is a ℤ_{p}- root datum. We define a cover of D to be any ℤ_{p}-root datum of this form. In particular the universal cover D̃ of D is defined by D̃ = (W, L_{0}, {ℤ_{p}b_{σ}}). Note that by definition, π_{1}(D̃) = 0. For the reduction in Section 2 we need the following result which does not rely on the fundamental group formula Theorem 8.6.

Proposition 8.9. Let X be a connected p-compact group with ℤ_{p}-root datum D_{X} and let H be a subgroup of π_{1}(X). Let Y → X be the cover of X corresponding to H. Then D_{Y} is the cover of D_{X} corresponding to the kernel of the composition L_{X} → π_{1}(D_{X}) → π_{1}(X) → π_{1}(X)/H.

Proof. By construction BY is the fiber of BX → B^{2}(π_{1}(X)/H). Let BT → BX be a maximal torus of X and let BN_{X} → BX the maximal torus normalizer. Now consider the following diagram obtained by pulling the fibration BY → BX → B^{2}(π_{1}(X)/H) back along BT → BN_{X} → BX

(8.3)  [figure: BT′; BN′; BY; BT; BN_{X}; BX; B^{2}(π_{1}(X)/H); B^{2}(π_{1}(X)/H); B^{2}(π_{1}(X)/H).]

Thus BT′ → BY is a maximal torus and BN′ → BY is a maximal torus normalizer by [44, Thm. 1.2]. The above diagram shows that the Weyl group of Y identifies with the Weyl group W of X. For a reflection σ ∈ W we have the diagram

\xymatrix{BT̆′ \ar[r] \ar[d] & BN̆′(σ) \ar[d] \ar[r] & BY (σ) \ar[d] \\ BT̆ \ar[r] & BN̆_{X}(σ) \ar[r] & BX(σ),}

where X(σ) = C_{X}(T̆_{0}^{+}(σ)), N̆_{X}(σ) = CN̆_{X} (T̆_{0}^{+}(σ)) and similarly for Y (σ) and N̆′(σ). It now follows by definition (cf. [28, §9]) that the image of h′_{σ} ∈ T̆′ equals h_{σ} ∈ T̆. Diagram (8.3) produces the short exact sequence 0 → L_{Y} → L_{X} → π_{1}(X)/H → 0 so ℤ_{p}b′_{σ} ⊆ L_{Y} maps to ℤ_{p}b_{σ} ⊆ L_{X}. This shows the claim. □

We next introduce quotients of root data. Let D = (W, T̆, {h_{σ}}) be a ℤ_{p}-root datum, and A ⊆ Z̆(D) a subgroup of the discrete center. We define a quotient of D to be a ℤ_{p}-root datum of the form D/A = (W, T̆/A, \{\overline{h_{σ}}\}) where \overline{h_{σ}} denotes the image of h_{σ} in T̆/A; the fact that this is a ℤ_{p}-root datum is part of the following result.

Proposition 8.10. (1) If D = (W, T̆, {h_{σ}}) is a ℤ_{p}-root datum and A ⊆ Z̆(D), then D/A = (W, T̆/A, \{\overline{h_{σ}}\}) is a ℤ_{p}-root datum. Moreover { \atop D/A} ≅ D̃ and Z̆(D/A) ≅ Z̆(D)/A.

(2) Let X be a connected p-compact group with ℤ_{p}-root datum D_{X} and A → X a central monomorphism. If Ă denotes the discrete approximation to A, then the ℤ_{p}- root datum D_{X/A} of the p-compact group X/A identifies with the quotient datum

D_{X}/Ă. In particular { \atop D_{X/A}} ≅ D̃_{X}.

Proof. Write D = (W, L, {ℤ_{p}b_{σ}}) where L = Hom_{ℤ_{p}}(ℤ/p^{∞}, T̆) is the associated ℤ_{p}-lattice and b_{σ} ∈ L the associated coroots. By (8.2) the sequence of discrete tori T̆ → T̆/A → T̆/Z̆(D) corresponds to the sequence L → L_{T̆/A} → M_{0}^{∗} of ℤ_{p}-lattices, where M_{0} is the root lattice spanned by the β_{σ}. Note that (W, L^{∗}) is a reflection group via the action σ(α) = α + α(b_{σ^{−1}})β_{σ^{−1}} so (W, L^{∗}, {ℤ_{p}β_{σ^{−1}}}) is a ℤ_{p}-root datum (the dual of D). Applying Lemma 8.8 to this ℤ_{p}-root datum shows that M_{0} ⊕ (L^{∗})^{W} has finite index in L^{∗}, so the

dual homomorphism L → M_{0}^{∗} ⊕ ((L^{∗})^{W} )^{∗} is injective. The last summand is isomorphic to Hom_{ℤ_{p}}(H_{0}(W; L), ℤ_{p})^{∗} which again identifies with H_{0}(W; L) modulo its torsion subgroup. The formula σ(x) − x = β_{σ}(x)b_{σ} shows that the image of the composition L_{0} → L → H_{0}(W; L) is torsion, so by the above the composition L_{0} → L → L_{T̆/A} → M_{0}^{∗} is injective. In

particular W acts faithfully on L_{T̆/A} by Lemma 8.8. The homomorphism β_{σ}⊗_{ℤ_{p}}ℤ/p^{∞} : T̆ →

ℤ/p^{∞} factors through T̆/A, and we get a corresponding homomorphism β′_{σ} : L_{T̆/A} → ℤ_{p}

such that the composition L → L_{T̆/A} ⟶^{β′_{σ}} ℤ_{p} agrees with β_{σ}. We now claim that σ(x) = x+β′_{σ}(x)b_{σ} for x ∈ L_{T̆/A}. To see this note that the image of L in L_{T̆/A} has finite index since we have the exact sequence L → L_{T̆/A} → Ext_{ℤ_{p}}^{1}(ℤ/p^{∞}, A) with Ext_{ℤ_{p}}^{1}(ℤ/p^{∞}, A) finite. Thus the claim follows from the above by using the corresponding formula σ(x) = x + β_{σ}(x)b_{σ} for x ∈ L. This proves that D/A = (W, L_{T̆/A}, {ℤ_{p}b_{σ}}) is a ℤ_{p}-root datum. In particular we

obtain { \atop D/A} = (W, L_{0}, \{ℤ_{p}b_{σ}\}) = D̃ by definition.

To see the claim about Z̆(D/A), note that by the above β_{σ} ⊗_{ℤ_{p}} ℤ/p^{∞} : T̆ → ℤ/p^{∞}

identifies with the composition T̆ → T̆/A ⟶^{β′_{σ}⊗_{ℤp}ℤ/p^{∞}} ℤ/p^{∞}. Hence the singular sets for D and D/A satisfies S_{D}(σ)/A = S_{D/A}(σ) and the claim follows.

To see part (2), let i : BT → BX be a maximal torus and let f : BA → BX be the central monomorphism. Then f factors through BT by [25, Prop. 8.11] to give a central monomorphism g : BA → BT. Moreover i factors through the maximal torus normalizer N of X and we obtain the diagram

\xymatrix{BT \ar[r] \ar[d] & BN \ar[r] \ar[d] & BX \ar[d] \\ BT/A \ar[r] & BN /A \ar[r] & BX/A}

cf. Construction 2.2. It follows that T/A is a maximal torus in X/A and N /A is the maximal torus normalizer. The Weyl groups of X and X/A identifies naturally, cf. [45, Thm. 4.6]. By construction the elements h′_{σ} ∈ T̆/Ă corresponding to D_{X/A} are the images of the elements h_{σ} ∈ T̆ corresponding to X (cf. the proof of Proposition 8.9). This shows that D_{X/A} ≅ D_{X}/Ă as desired. □

As a special case of the quotient construction, we define the adjoint D_{ad} of a ℤ_{p}-root datum D = (W, L, {ℤ_{p}b_{σ}}) by D_{ad} = D/Z̆(D). Note that by Proposition 8.10(1) we have Z̆(D_{ad}) = 0 and that it follows from the proof that D_{ad} = (W, M_{0}^{∗}, {ℤ_{p}b_{σ}}), where M_{0} is the root lattice, i.e., the sublattice of L^{∗} spanned by the roots β_{σ}.

Proposition 8.11. Any ℤ_{p}-root datum with Z̆(D) = 0 or π_{1}(D) = 0 splits as a direct product D ≅ D_{1} × … × D_{n} of irreducible ℤ_{p}-root data D_{i}.

Proof. The case Z̆(D) = 0 is essentially proved by Dwyer-Wilkerson [27, Pf. of Thm. 1.5], for completeness we briefly sketch the argument: Let (W, L) be the ℤ_{p}-reflection group associated to D and write L ⊗_{ℤ_{p}} ℚ_{p} = V_{1} ⊕ … ⊕ V_{n} be the decomposition of L ⊗_{ℤ_{p}} ℚ_{p} into irreducible ℚ_{p}[W]-modules. Define L_{i} = L ∩ V_{i}. Since Z̆(D) = 0 we have ⋂_{σ} T̆_{0}^{+}(σ) = 0 as well, and it follows [27, Pf. of Thm. 1.5] that the homomorphism L_{1} × … × L_{n} → L is an isomorphism. Letting W_{i} denote the point-wise stabilizer of L_{1} ⊕ … ⊕ L̂_{i} ⊕ … ⊕ L_{n} we hence (cf. [27, Prop. 7.1]) get a product decomposition (W, L) ≅ (W_{1}, L_{1}) × … × (W_{n}, L_{n}). It is now clear that there is a unique ℤ_{p}-root datum structure D_{i} on (W_{i}, L_{i}) such that we get a product decomposition D ≅ D_{1} × … × D_{n} into irreducible ℤ_{p}-root data.

The case where π_{1}(D) = 0 is easily reduced to the first case using the previous results: By Proposition 8.10(1), Z̆(D_{ad}) = 0 so we can write D_{ad} ≅ D_{1} × … × D_{n} where the D_{i} are irreducible. Proposition 8.10(1) now shows that D = D̃ ≅ D̃_{ad} ≅ D̃_{1} × … × D̃_{n} as claimed. □

Theorem 8.12 (The classification of ℤ_{p}-root data; structure version). (1) Let D = (W, L, {ℤ_{p}b_{σ}}) be a ℤ_{p}-root datum with coroot lattice L_{0}, and let D′ = (W, L_{0} ⊕ L^{W} , {ℤ_{p}b_{σ}}) = D̃ × D_{triv}, where D_{triv} = (1, L^{W} , ∅) is a trivial ℤ_{p}-root datum. Then D ≅ D′/A for a finite central subgroup A ⊆ Z̆(D′) and there is a splitting D̃ ≅ D_{1} × … × D_{n} of D̃ into irreducible ℤ_{p}-root data D_{i} with π_{1}(D_{i}) = 0.

(2) For p > 2, the assignment D = (W, L, {ℤ_{p}b_{σ}}) ⇝ (W, L ⊗_{ℤ_{p}} ℚ_{p}) is a one-to-one correspondence between isomorphism classes of irreducible ℤ_{p}-root data D with π_{1}(D) = 0 and isomorphism classes of irreducible ℚ_{p}-reflection groups. For p = 2, the assignment is surjective and the preimage of every element consists of a single element, except for (W_{Sp(n)}, L_{Sp(n)} ⊗ ℚ_{2}) ≅ (W_{Spin(2n+1)}, L_{Spin(2n+1)} ⊗ ℚ_{2}) whose preimage consists of D_{Sp(n)} ⊗_{ℤ} ℤ_{2} and D_{Spin(2n+1)} ⊗_{ℤ} ℤ_{2} which are non-isomorphic for n ≥ 3.

Proof. The fact that D′ is a ℤ_{p}-root datum follows from Lemma 8.8. The short exact sequence 0 → L_{0} ⊕ L^{W} → L → F → 0 where F is finite with trivial W -action produces a short exact sequence 1 → A → T̆′ → T̆ → 1 between the associated discrete tori. Since the roots β′_{σ} for D′ are given by L_{0} ⊕ L^{W} → L ⟶^{β_{σ}} ℤ_{p}, it follows that A ⊆ S_{D′}(σ) = ker((L_{0} ⊕ L^{W}) ⊗_{ℤ_{p}} ℤ/p^{∞} → ℤ/p^{∞}) for any reflection σ. Hence A ⊆ Z̆(D′) is central and D ≅ D′/A. The last part of (1) follows from Proposition 8.11.

We now prove (2). For any prime p, Theorem 8.1(2) shows that the assignment in (2) gives a one-to-one correspondence between isomorphism classes of exotic ℤ_{p}-root data and isomorphism classes of exotic ℚ_{p}-reflection groups, and that π_{1}(D) = 0 for all exotic ℤ_{p}-root data D. Hence it suffices by Theorem 8.1(1) to show the claim for ℤ_{p}-root data of the form D_{1} ⊗_{ℤ} ℤ_{p} for a ℤ-root datum D_{1}.

In this case it is clear that if π_{1}(D_{1} ⊗_{ℤ} ℤ_{p}) = 0, then we can find a ℤ-root datum D′_{1} with π_{1}(D′_{1}) = 0 and D′_{1} ⊗_{ℤ} ℤ_{p} ≅ D_{1} ⊗_{ℤ} ℤ_{p} (simply choose D′_{1} to be the universal cover of D_{1}; this is defined for ℤ-root data in the same way as for ℤ_{p}-root data). Hence it suffices to study the assignment D = (W, L, {ℤ_{p}b_{σ}}) ⇝ (W, L ⊗_{ℤ} ℚ) from irreducible ℤ-root data with π_{1}(D) = 0 to irreducible ℚ-reflection groups. It is well-known (cf. [9, §4]) that this assignment is surjective and that it only fails to be injective in that the ℤ-root data D_{Sp(n)} and D_{Spin(2n+1)} which are non-isomorphic for n ≥ 3 maps to the same ℚ-reflection group. This proves part (2) since for n ≥ 3, the ℤ_{p}-root data D_{Sp(n)} ⊗_{ℤ} ℤ_{p} and D_{Spin(2n+1)} ⊗_{ℤ} ℤ_{p} are non-isomorphic for p = 2 and isomorphic for p = 2. □

8.4. Automorphisms. Recall that an isomorphism between two ℤ_{p}-root data D = (W, L, {ℤ_{p}b_{σ}}) and D′ = (W′, L′, {ℤ_{p}b′_{σ′}}) is an isomorphism φ : L → L′ with the property that φW φ^{−1} = W′ as subgroups of Aut(L′) and φ(ℤ_{p}b_{σ}) = ℤ_{p}b′_{φσφ^{−1}} for every reflection σ ∈ W. We denote the automorphism group of D by Aut(D); clearly W is a normal subgroup of Aut(D), and we define the outer automorphism group Out(D) by Out(D) = Aut(D)/W.

Recall that a ℤ_{p}-root datum D = (W, L, {ℤ_{p}b_{σ}}) is called irreducible if L ⊗_{ℤ_{p}} ℚ_{p} is an irreducible ℚ_{p}[W]-module. The following proposition is a restatement of [8, Prop. 5.4].

Proposition 8.13. Suppose D_{i} = (W_{i}, L_{i}, {ℤ_{p}b_{σ}}_{σ∈Σ_{i}}), i = 0, …, k, is a collection of pairwise non-isomorphic irreducible ℤ_{p}-root data. Assume that W_{0} = 1 but that W_{i} is nontrivial for i ≥ 1. Let D = ∏_{i=0}^{k} D_{i}^{m_{i}}, m_{i} ≥ 1 denote a product of these ℤ_{p}-root data. Then

GL_{m_{0}}(ℤ_{p}) × (∏_{i=1}^{k} Out(D_{i}) ≀ Σ_{m_{i}}) ⟶^{≅} Out(D).

Proof. This follows directly by combining [8, Prop. 5.4] with [4, Rem. 4.5]. □

The following two results are needed for the reduction in Section 2.

Proposition 8.14. Let D = (W, L, {ℤ_{p}b_{σ}}) be a ℤ_{p}-root datum with coroot lattice L_{0}. Let D′ = (W, L_{0} ⊕L^{W} , {ℤ_{p}b_{σ}}) = D̃×D_{triv}, where D_{triv} = (1, L^{W} , ∅) is a trivial ℤ_{p}-root datum.

Then D ≅ D′/A for a finite subgroup A ⊆ Z̆(D′) and the restriction

Aut(D) ⟶ Aut(D′) = Aut(D̃) × Aut(D_{triv})

is an isomorphism onto the subgroup {φ ∈ Aut(D′) | φ(A) = A}. In particular Out(D) identifies with a subgroup of finite index in Out(D′).

Remark 8.15. For a ℤ_{p}-root datum D = (W, L, {ℤ_{p}b_{σ}}) with π_{1}(D) = 0 we have ℤ_{p}b_{σ} = ker(L ⟶^{N} L), where N = 1 + σ + … + σ^{|σ|−1} is the norm element: By Theorem 8.1(1) either D ≅ D_{1} ⊗_{ℤ} ℤ_{p} for a ℤ-root datum D_{1} with π_{1}(D_{1}) = 0 (cf. the proof of Theorem 8.12(2)) or D is exotic. In the first case the result is well known [9, §4], and in the second case the claim follows since H^{1}(⟨σ⟩; L) = 0 for all reflections σ (cf. the proof of Theorem 8.1(2)). In particular we obtain Out(D) = N_{GL(L)}(W)/W, and this group is explicitly computed for all irreducible (W, L) in [8, Thm. 13.1].

Proof of Proposition 8.14. The fact that D ≅ D′/A for a finite subgroup A ⊆ Z̆(D′) is part of Theorem 8.12(1), and the identification Aut(D′) = Aut(D̃) × Aut(D_{triv}) comes from Proposition 8.13. Clearly both L_{0} and L^{W} are invariant under Aut(D) so we get a restriction homomorphism Aut(D) ⟶ Aut(D′) which is injective since L_{0} ⊕ L^{W} has finite index in L by Lemma 8.8. If φ ∈ Aut(D), then φ corresponds to an automorphism φ : T̆ → T̆ and the restriction of φ to L_{0} ⊕ L^{W} corresponds to a lift φ̃ : T̆′ → T̆′ which sends A to itself. Conversely, if φ ∈ Aut(D′) with φ(A) = A, then φ clearly induces an automorphism of D. The last claim follows from the fact that the orbit of A under Out(D′) is finite since Z̆(D′) has only finitely many subgroups isomorphic to A. □

Corollary 8.16. For any ℤ_{p}-root datum D there is a canonical isomorphism

Aut(D̃) ⟷^{≅} Aut(D_{ad}).

Proof. Write D = (W, L, {ℤ_{p}b_{σ}}), and let M_{0} ⊆ L^{∗} denote the root lattice, i.e., the lattice spanned by the roots β_{σ}. Then D̃ = (W, L_{0}, {ℤ_{p}b_{σ}}) and the roots for D̃ are given by

the composition L_{0} → L ⟶^{β_{σ}} ℤ_{p}. Hence we can identify the root lattice for D̃ with M_{0} and hence D̃/Z̆(D̃) = (D̃)_{ad} ≅ (W, M_{0}^{∗}, {ℤ_{p}b_{σ}}) = D_{ad}. The result now follows from Proposition 8.14. □

8.5. Finiteness properties. In this final subsection we prove that there are only finitely many ℤ_{p}-root data of a given rank, and that for a fixed ℤ_{p}-root datum D, Out(D) only contains finitely many finite subgroups up to conjugation. These results are used for the proof of the finiteness statements in Theorems 1.5 and 1.6 in Section 7.

Proposition 8.17. For any prime p there is, up to isomorphism, only finitely many ℤ_{p}-root data of a fixed rank.

Proof. By the classification of finite ℚ_{p}-reflection groups [13] [22], there are only finitely many finite ℚ_{p}-reflection groups of a fixed rank. For each finite ℚ_{p}-reflection group (W, V), there are, up to equivalence, only finitely many representations W → GL(V) which gives rise to the reflection group (W, V). For each of these representations, the local version of the Jordan-Zassenhaus theorem [14, Thm. 24.7] shows that, up to isomorphism, there are only finitely many ℤ_{p}[W]-lattices L with L⊗_{ℤ_{p}} ℚ_{p} ≅ V as ℚ_{p}[W]-modules. In particular we conclude that, up to isomorphism, there are only finitely many finite ℤ_{p}-reflection groups of fixed rank. Finally, choosing a ℤ_{p}-root datum for a finite ℤ_{p}-reflection group (W, L) corresponds to choosing a cyclic subgroup of the finite group H^{1}(⟨σ⟩; L) for each conjugacy class of reflections σ. Hence any finite ℤ_{p}-reflection group gives rise to only finitely many ℤ_{p}-root data. This proves the result. □

Proposition 8.18. Let D be a ℤ_{p}-root datum. Then Out(D) contains only finitely many conjugacy classes of finite subgroups.

Proof. Let (W, L) be the finite ℤ_{p}-reflection group underlying D, and let n = rank L. Note first that the order of a finite subgroup of GL(L) is bounded above: If G ⊆ GL_{n}(ℤ_{p}) has finite order, then it is easily seen that the composition G → GL_{n}(ℤ_{p}) → GL_{n}(𝔽_{p}) is injective for p > 2 and has kernel of order at most 2^{n^{2}} for p = 2 (cf. [8, Lem. 11.3]). Hence G has order at most |GL_{n}(𝔽_{p})| for p > 2 and 2^{n^{2}} · |GL_{n}(𝔽_{2})| for p = 2. In particular there is an upper bound on the order of finite subgroups of N_{GL(L)}(W). Since Out(D) is contained in N_{GL(L)}(W)/W and W is finite, it follows that there is also an upper bound on the order of finite subgroups of Out(D).

Fix a finite group G. By the above it suffices to show that the set Rep(G, Out(D)) (i.e., the set of homomorphisms G → Out(D) modulo conjugation in Out(D)) is finite. By Theorem 8.12(1), we can write D ≅ D′/A where D′ = D̃ × D_{triv} and D_{triv} is a trivial ℤ_{p}-root datum, and this identifies Out(D) with a subgroup of finite index in Out(D′), cf. Proposition 8.14. Hence it is enough to prove that Rep(G, Out(D′)) is finite since this will imply that Rep(G, Out(D)) is finite.

By Theorem 8.12(1) and Proposition 8.13, Out(D′) is isomorphic to a direct product of GL_{m_{0}}(ℤ_{p}) and groups of the form Out(D_{i}) ≀ Σ_{m_{i}} where the D_{i} are irreducible ℤ_{p}-root data. It thus suffices to know that Rep(G, GL_{m}(ℤ_{p})) and Rep(G, Out(D) ≀ Σ_{m}) are finite for any m and any irreducible ℤ_{p}-root datum D. The first claim follows directly from the local version of the Jordan-Zassenhaus theorem [14, Thm. 24.7].

To see the second claim, let W denote the Weyl group of D and note that since D is irreducible, Schur’s lemma implies that the image of the central homomorphism ℤ_{p}^{×} → Out(D) equals the kernel of the canonical homomorphism Out(D) → Out(W). Since ℤ_{p}^{×} ≅ ℤ_{p} × C where C is finite (C = ℤ/2 for p = 2 and C = ℤ/(p − 1) for p > 2) and Out(W) is finite, it follows that Out(D) is a finite (central) extension of ℤ_{p}. In particular E = Out(D) ≀ Σ_{m} fits into an extension of the form

(8.4)  1 → (ℤ_{p})^{m} → E ⟶^{π} Q → 1,

where Q is finite. Any homomorphism φ : G → E gives a homomorphism π ∘ φ : G → Q by composition, and for a fixed homomorphism α : G → Q, the set of homomorphisms φ : G → E with π∘φ = α equals the set of splittings of the pullback 1 → (ℤ_{p})^{m} → E′ → G → 1 of (8.4) along α : G → Q. The set of such splittings, modulo conjugation by elements in (ℤ_{p})^{m}, is in one-to-one correspondence with H^{1}(G; (ℤ_{p})^{m}), which is finite for all actions of G on (ℤ_{p})^{m}. Hence the set of homomorphisms φ : G → E with π ∘ φ = α is finite modulo conjugation in (ℤ_{p})^{m} ⊆ E. Since Q is finite, there are only finitely many homomorphisms α : G → Q, so we conclude that Rep(G, E) is finite as claimed. □

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Institute of Mathematical Sciences, Aarhus University

E-mail address: kksa@math.ku.dk

Dept. of Mathematics, University of Chicago, and Dept. of Math. Sci., Univ. of Copenhagen E-mail address: jg@math.ku.dk